← Notes on advances of LSC
factorisation systems__20250707.tex
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\title{local state classifier relative to factorization system}
\author{Ryuya Hora}
\thanks{ZEN University. \url{ryuya_hora@zen.ac.jp}}
% \date{\today}
\subjclass[2020]{MSC}
\keywords{Keywords}
\begin{document}
\begin{abstract}
The notion of a local state classifier is defined to be a colimit of all monomorphisms. This note aims to prove a general existence theorem for such a large colimit. First, we generalize the notion of a local state classifier to general orthogonal factorization system (not only the epi-mono factorization system in a topos). Then, we prove that the generalized local state classifier relative to a suitable factorization system on an arbitrary total category exists including all locally presentable categories.
\end{abstract}
\maketitle
\tableofcontents
% \section{Quotient maps}
\section{Introduction:
% }
% \subsection{
Local state classifier and its construction problem}
In \cite{hora2024internal}, the notion of a local state classifier was introduced as follows:
\begin{definition}[\cite{hora2024internal}]
A \demph{local state classifier} $\Xi$ of a category $\C$ is the colimit of all monomorphisms
\[
\Xi\coloneqq \colim (\C_{\mathrm{mono}}\rightarrowtail \C)
\]
if it exists.
\end{definition}
Even if $\C$ is (small) cocomplete, $\C$ might not admit a local state classifier, since the indexing category $\C_{\mathrm{mono}}$ is not small. In the same paper \cite{hora2024internal}, the author proved that any Grothendieck topoi admits a local state classifier by concretely constructing it.
\begin{proposition}[\cite{hora2024internal}]
For a small site $(\C,J)$, the local state classifier $\Xi$ of the sheaf topos $\Sh(\C, J)$ is given by the sheafification $\Xi=\a\Xi_0$ of the following presheaf:
\[
\Xi_0 \colon c \mapsto \{\text{quotient objects of }\a\yo(c)\}.,
\]
where $\a\colon \PSh(\C) \to \Sh(\C,J)$ denotes the sheafification functor and $\yo\colon \C \to \PSh(\C)$ denotes the yoneda embedding.
\end{proposition}
However, this construction is not fully satisfactory, because it is too technical and does not provide any conceptual understanding. The aim of this paper is to demystify the construction
% of a local state classifier
by clarifying connections with two classical categorical notions:
\begin{itemize}
\item total categories, and
\item orthogonal factorization systems.
\end{itemize}
From the next section, we will see them one by one.
\section{Total categories}
This section aims to recall the definition and basic properties of total categories.
\subsection{Definition and examples}
\begin{definition}\memo{See e.g. \cite{street1978yoneda, tholen1980note, street1984family}}
A locally small category $\C$ is said to be \demph{total} if its Yoneda embedding $\C \xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$ admits a left adjoint $L\colon \Func{\C^{\op}}{\Set} \to \C$.
\end{definition}
In this note, total category means a locally small total category.
\begin{example}[Easiest example]
The easiest category that one can check the totality is the terminal category $\1$. The yoneda embedding
\[
\yo_{\1}\colon \1 \hookrightarrow \Set
\]
admits a left adjoint, which is given by the unique functor $\Set \to \1$.
% The category of sets is \cite{rosebrugh1994adjoint}
\end{example}
\begin{example}[Complete lattice]
Let $P$ be a small complete lattice. Then, we can prove that $P$ is total by showing that the functor
\[
L\colon\PSh(P) \to P \colon F \mapsto \sup\{p\in P\mid F(p) \neq \emptyset\}
\]
is a left adjoint to the yoneda embedding.
\end{example}
\begin{example}[Presheaf categories]
The first example of a total and large category is a presheaf category. Let $J$ be an arbitrary small category. Then, one can prove that the pre-composition functor ${-}\circ\yo_{J} \colon \PSh(\PSh(J)) \to \PSh(J)$ is a left adjoint to the Yoneda embedding functor $\yo_{\PSh(J)} \colon \PSh(J) \hookrightarrow \PSh(\PSh(J))$. (In the case $J=\1$, this is a part of the adjoint $5$-tuple in \cite{rosebrugh1994adjoint}.)
\end{example}
\begin{example}
\memo{cite} It is known that any reflective subcategory of a total category is total, and that any presheaf category is total. Therefore, all locally presentable categories, including all Grothendieck topoi and all algebraic categories, are total. Other examples include $\Top$.
\end{example}
\subsection{Small and large colimits in a total category}
Since a (possibly large) colimit in a reflective subcategory is given by the reflector functor, the calculation of a colimit of $J \xrightarrow{F} \C$ is reduced to that of $J \xrightarrow{F} \C \xrightarrow{\yo_\C} \PSh(\C)$ for a total category $\C$.
\begin{lemma}\label{lem:colimitAfterYoneda}
For any locally small category $\C$ and any (possibly large) diagram $F \colon J \to \C$, the colimit of
% $\yo_{\C}\circ F \colon J \to \C \to \PSh(\C)$
$J \xrightarrow{F} \C \xrightarrow{\yo_\C} \PSh(\C)$
is given by the presheaf
\[
\pi_{0}({-}\downarrow F)\colon c \mapsto \pi_0(c\downarrow F)
\]
if $\pi_0(c\downarrow F)$ is small for each $c\in \C$.
\end{lemma}
\begin{proof}
A colimit in a functor category is given by objectwise colimits as soon as the objectwise colimits exist. Therefore, our problem is reduced to the calculation of the colimit of the functor
% Since the functor $J \xrightarrow{F} \C \xrightarrow{\yo_\C} \PSh(\C)$ is equal to
\[
\begin{tikzcd}
J\ar[r, "F"] &\C \ar[r, "{\yo_{\C}}"] \ar[rr, bend right, "{\C(c,-)}"']& \PSh(\C)\ar[r, "\ev_c"] &\Set.
\end{tikzcd}
\]
In general, the colimit of a functor $G\colon J \to \Set$ is given by $\pi_0\left(\int G\right)$ if it is small. This completes the proof since $\int \C(c,-)\circ F = c\downarrow F$.
\end{proof}
\begin{example}[Final functors and the colimit of yoneda]
If $F\colon J \to \C$ is final (i.e., $\pi_0(c\downarrow F)$ is singleton), then the colimit of $J \xrightarrow{F} \C \xrightarrow{\yo_\C} \PSh(\C)$ is given by the terminal presheaf. In particular, if $F= \id_{\C}\colon J=\C \to \C$, the colimit of the yoneda embedding is the terminal presheaf.
\end{example}
\begin{proposition}[colimits in total categories]
Any total category is small cocomplete. Furthermore,
For any total category $\C$ with $L \dashv \yo_{\C}$ and any (possibly large) diagram $F\colon J \to \C$, if
\[
\C(c, F{-})\colon J \xrightarrow{F} \C \xrightarrow{\C(c,-)}\Set
\]
admits a colimit for any $c\in \C$, then there exists a colimit of $F\colon J \to \C$, which is given by
\[
\colim_{j\in J} Fj = L\left( \colim_{j\in J} \C(c, F{j})\right).
\]
In particular, if the comma category $c \downarrow F$ has only small number of connected components for each $c\in \C$, then $F$ admits a colimit.
\[
\colim_{J} F = L\left(\pi_0(c\downarrow F)\right).
\]
\end{proposition}
\section{Orthogonal factorization system}
\subsection{Definition and examples}
This subsection aims to recall the basic properties of orthogonal factorization systems.
\begin{definition}\label{def:orthogonalfs}
An \demph{orthogonal factorization system} on a category $\C$ is a pair $(\EC, \MC)$ of subclasses $\EC, \MC \subset \Mor (\C)$ that satisfies the following conditions.
\begin{itemize}
\item Both $\EC, \MC$ contains all isomorphisms.
\item Both $\EC, \MC$ are closed under compositions.
\item Any morphism in $\C$ is uniquely decomposed into a morphism in $\EC$ followed by one in $\MC$ up to unique isomorphisms.
\end{itemize}
\end{definition}
\begin{example}[Trivial example]
For any category $\C$, the pair $(\EC=\Mor, \MC=\Iso)$ is an orthogonal factorization system. Dually, the pair $(\EC=\Iso, \MC=\Mor)$ is also an orthogonal factorization system.
\end{example}
\begin{example}[Regular categories, topoi, abelian categories, and categories of algebras]
For any regular category $\C$, the pair $(\EC=\text{regular epi}, \MC= \text{mono})$ is an orthogonal factorization system. In particular,
\begin{itemize}
\item for any elementary topos $\C$, we have epi-mono factorization system,
\item for any abelian category $\C$, we have epi-mono factorization system, and
\item for any category of models of an algebraic theory $\C$, we have $(\text{Surj. hom.}, \text{inj. hom.})$-factorization system.
\end{itemize}
\end{example}
\begin{example}[quasitopoi]
\end{example}
\begin{example}[Locally presentable categories.]
For any locally presentable category $\C$, the pair $(\EC=\text{strong epi}, \MC= \text{mono})$ is an orthogonal factorization system. (\cite{adamek1994locally})
\end{example}
\begin{example}[Graphical example]\label{exmp:GraphicalExampleOfOFS}
Let $\R^2$ denote the poset (since a category) whose objects are elements of $\R^2$ and morphisms are inequalities in the product poset $(\R, \leq )^2$. Then the following classes
\begin{itemize}
\item $\EC=\{(x,y) \leq (x',y')\mid x=x'\}$\text{: vertical ones}
\item $\MC=\{(x,y) \leq (x',y')\mid y=y'\}$ \text{: horizontal ones}
\end{itemize}
define an orthogonal factorization system on the category $\R^2$.
Trivially, both classes contain all isomorphisms, which are all identities, and closed under compositions. The unique decomposition of $f: (x,y)\leq (x',y')$ is given by $(x,y) \underset{\EC}{\leq} (x,y')\underset{\MC}{\leq} (x',y')$ (\Cref{fig:DecompostionOFS}).
\begin{figure}[htbp]\label{fig:DecompostionOFS}
\centering
\begin{tikzpicture}[scale=0.7]
% Grid
\draw[very thin, gray] (-3.5,-3.5) grid (3.5,3.5);
% Axes
\draw[->, thick] (-4,0) -- (4,0) node[anchor=west]{$x$};
\draw[->, thick] (0,-4) -- (0,4) node[anchor=south]{$y$};
\draw[->, thick, black] (-3,-2) -- (3,3) node[anchor=west]{};
\draw[->, thick, blue] (-3,-2) -- (-3,3) node[anchor=east]{$\EC$};
\draw[->, thick, red] (-3,3) -- (3,3) node[anchor=west]{$\MC$};
\end{tikzpicture}
\end{figure}
% \begin{figure}[htbp]
% \centering
% \begin{tikzpicture}[scale=0.7]
% % Grid
% \draw[very thin, gray] (-3.5,-3.5) grid (3.5,3.5);
% % Axes
% \draw[->, thick] (-4,0) -- (4,0) node[anchor=west]{$x$};
% \draw[->, thick] (0,-4) -- (0,4) node[anchor=south]{$y$};
% % Horizontal arrow (on the line y=2)
% \draw[->, thick, blue] (-3,-2) -- (1,-2) node[anchor=west]{};
% % Vertical arrow (on the line x=-2)
% \draw[->, thick, red] (3,-1) -- (3,3) node[anchor=south]{};
% \end{tikzpicture}
% \end{figure}
\end{example}
\subsection{Local state classifier relative to factorization system}
\begin{notation}
For an orthogonal factorization systems $(\EC,\MC)$ on a category $\C$, the wide subcategory of $\C$ consisting of all morphisms in $\EC$ (resp. $\MC$) is denoted by $\C_\EC$ (resp. $\C_\MC$).
\end{notation}
\begin{definition}
For a category $\C$ equipped with an orthogonal factorization systems $(\EC,\MC)$, the \demph{$\MC$-local state classifier} is the colimit of the embedding functor $\C_\MC \rightarrowtail \C$, if it exists.
\end{definition}
\begin{example}
If $\MC$ is the class of all monomorphisms, then the $\MC$-local state classifier is the local state classifier in \cite{hora2024internal}.
\end{example}
\begin{example}
For any category $\C$, the pair $(\EC=\Iso, \MC=\Mor)$ is an orthogonal factorization system.
The $\Mor$-local state classifier is the colimit of the identity functor, which is known to be the same as the terminal object of $\C$ (See \cite{riehl2017category} and \cite{menni2025nonsingular}). Therefore, a category $\C$ admits a $\Mor$-local state classifier if and only if $\C$ admits a terminal object.
\memo{This example is not boring. The monoid structure induced on the terminal object might be useful in game theory or in other coalgebraic contexts.}
\end{example}
\subsection{The presheaf of quotient objects and its univesality}
\begin{definition}
Let $(\EC,\MC)$ be an orthogonal factorization system on a category $\C$.
For each $c\in \ob(\E)$, let $\QuoE(c)$ denote the class of all isomorphism classes of $c/\C_{\EC}$.
% morphisms in $\EC$ from the object $c$ regarded as objects in $c/\C$.
\end{definition}
For any morphism $f\colon c \to d$ in $\C$, we define $q_f\in \QuoE(c)$ as (the isomorphism class of) the $\EC$-part of its $(\EC,\MC)$-factorization. This is well-defined since the factorization is unique up to a (unique) isomorphism.
\begin{notation}
Assuming that $\C$ is \demph{$\EC$-cowell-powered}, which means that $\QuoE(c)$ is small for any $c\in \ob(\C)$, we define the functor
\[
\QuoE\colon \E^{\op} \to \Set
\]
by the factorization (\memo{write it later}).
% \[
% \QuoE(f\colon X \to Y)(e\colon Y \twoheadrightarrow Q) =
% \]
\end{notation}
What can we say about this presheaf $\QuoE\colon \C^{\op} \to \Set$?
% This functor $\QuoE$ is rarely representable:
\begin{remark}[$\QuoE$ is rarely representable]
In contrast to the situation that a $\MC$-subobject classifier represents the presheaf of subobjects (cf. topos, or quasitopos), the presheaf of quotient objects is rarely representble:
If $\QuoE$ is representable, then there exists an object $Q\in \E$ such that every object $X\in \ob(\E)$ admits an $\MC$-morphism into $Q$. \memo{Can we say $\MC=\C$?}
\end{remark}
\memo{If we do not assume properness, what is naturally defined might be a pseudo-functor $\E^{\op} \to \mathbf{CAT}$.}
\begin{remark}
\cite{kenney2006copower}
\end{remark}
But $\QuoE$ has a nice universal property, which is quite similar to that of a local state classifier:
\[
\QuoE = \colim(\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}).
\]
Since colimits in a functor category are calculated pointwise (if it exists in each point), we first study the functor
\[
\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set} \xrightarrow{\ev_c}\Set,
\]
which coincides with the covariant hom-functor $\C(c,-)$, but with the restricted domain $\MC$
\[
\MC \xrightarrow{\C(c,-)} \Set.
\]
% \begin{lemma}\label{lem:DecompositionIntoPresentables}\memo{false, we need to consider the groupoid action of $(X/\EC)_0$}
% The functor
% $
% \C(c,-)\colon \MC \to \Set
% $
% is decomposed into the (small) coproduct of representables as follows:
% \[
% \C(c,-) \cong \coprod_{[q\colon c \twoheadrightarrow d]\in \QuoE(c)} \MC(d,-).
% \]
% In particular, the functor $
% \C(c,-)\colon \MC \to \Set
% $ admits a colimit, which is given by the set $\QuoE(c)$ equipped with the colimit cocone
% \[
% \C(c, c') \to \Quo_E(c) \colon f \mapsto \text{(the $\EC$-part of the factorization of $f$)}.
% \]
% \end{lemma}
% \begin{proof}
% The isomorphism is nothing other than the unique existence of the $(\EC,\MC)$-factorization \memo{check maybe we need properness?}. The latter holds since the colimit of a representable functor is a singleton.
% \end{proof}
\begin{proposition}[Universality of $\Quo$]\memo{We can deduce the restricted versions from this} Assuming the local smallness of $\C$,
the presheaf $\QuoE\colon \C^{\op} \to \Set$ provides the colimit
\[
\QuoE = \colim(\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set})
\]
\end{proposition}
\begin{proof}
\memo{write}
\end{proof}
\begin{remark}If all morphisms in $\EC$ are epic,
the functor
$
\C(c,-)\colon \MC \to \Set
$
is decomposed into the (small) coproduct of representables as follows:
\[
\C(c,-) \cong \coprod_{[q\colon c \twoheadrightarrow d]\in \QuoE(c)} \MC(d,-).
\]
In particular, the functor $
\C(c,-)\colon \MC \to \Set
$ admits a colimit, which is given by the set $\QuoE(c)$ equipped with the colimit cocone
\[
\C(c, c') \to \Quo_E(c) \colon f \mapsto \text{(the $\EC$-part of the factorization of $f$)}.
\]
\begin{proof}
The isomorphism is nothing other than the unique existence of the $(\EC,\MC)$-factorization \memo{check maybe we need properness?}. The latter holds since the colimit of a representable functor is a singleton.
\end{proof}
\end{remark}
\begin{theorem}
% If a locally small category $\C$ with a factorization system $(\EC,\MC)$ satisfies the following conditions:
% \begin{itemize}
% \item $\C$ is total,
% \item $\C$ is $\EC$-cowell-powered,
% \end{itemize}
For any total category $\C$ with $\EC$-cowell powered factorization system $(\EC, \MC)$, $\C$ admits a $\MC$-local state classifier $\Xi_\MC$. Furthermore, the $\MC$-local state classifier $\Xi_\MC$ is given by
\[
\Xi_\MC = L(\QuoE),
\]
where $L\colon \Func{\C^{\op}}{\Set}\to \C$ denotes the left adjoint to the yoneda embedding.
\end{theorem}
Any locally presentable categories have LSC, due to the strong-epi mono factorization system.
\section{Classification of \texorpdfstring{$M$-}{}pullback stable family of \texorpdfstring{$M$-}{}subobjects.}
\memo{Consider $M$-subobject classifier}
\section{Examples}
\memo{nLab: A total category
is cartesian closed iff
L
preserves binary products (cf. Wood 1982, Thm. 9).}
\begin{example}[Grothendieck quasi-topoi]
\end{example}
\begin{example}[local state classifier in a locally presentable categories]
\end{example}
\appendix
\section{A direct proof of the universality of \texorpdfstring{$\QuoE$}{QuoE}}
In order to clarify the statement, let us write down the colimit cocone $\{\zeta_c \colon \yo(c) \to \QuoE\}$.
Each component $\zeta_{c,d} \colon \C(d,c) \to \QuoE (d)$ sends a morphism $f\colon d \to c$ to (the equivalence class of) the $\EC$-part of the factorization,
\[
\begin{tikzcd}
d \ar[rr, "f"] \ar[rd, twoheadrightarrow, "\zeta_{c,d}(f)"']&&c\\
&\bullet \ar[ru, "m_f"', rightarrowtail]&
\end{tikzcd}
\]
where $m_f$ denotes the $\MC$-part.
% Rigorously speaking, $\xi_{c,d}(f)$ is the equivalence class of the $\MC$-part.
\begin{lemma}
Each $\zeta_{c} \colon \yo(c) \to \QuoE$ is a natural transformation of the two presheaves $\yo(c)$ and $\QuoE$.
\end{lemma}
\begin{proof}
What we need to prove is that, for any composable maps
\[
d_1 \xrightarrow{g} d_0 \xrightarrow{f} c
\]
we have $\zeta_{c, d_1}(fg) = \zeta_{c, d_0}(f)*g$.
% \[
% \begin{tikzcd}
% d_1 \ar[r, "g"] \ar[d, twoheadrightarrow, "\zeta_{c, d_1}(fg)"']&d_0\ar[d, "\zeta_{c,d_1}(f)"', twoheadrightarrow]\\
% \bullet \ar[r, "\exists", rightarrowtail, dashed] & \bullet.
% \end{tikzcd}
% \]
% This follows since
% \[
% \begin{tikzcd}
% d_1 \ar[r, "g"] \ar[d, twoheadrightarrow, "\zeta_{c, d_1}(fg)"']&d_0\ar[d, "\zeta_{c,d_0}(f)"', twoheadrightarrow]\ar[dd, "f", bend left]\\
% \bullet \ar[r, "\exists", rightarrowtail, dashed]
% % \ar[rd, "m_{fg}"', rightarrowtail]
% & \bullet\ar[d, rightarrowtail, "m_f"']\\
% &c
% \end{tikzcd}
% \]
% we have
This follows since
\[
\begin{tikzcd}
d_1 \ar[r, "g"] \ar[d, twoheadrightarrow, "\zeta_{c, d_1}(fg)"']&d_0\ar[d, "\zeta_{c,d_1}(f)"', twoheadrightarrow]\\
\bullet \ar[r, "\exists", rightarrowtail, dashed] & \bullet.
\end{tikzcd}
\]
This follows since the following diagram proves that $\zeta_{c, d_0}(f)*g$ provides the $\EC$-part of the morphism $fg$.
\[
\begin{tikzcd}
d_1 \ar[r, "g"] \ar[d, twoheadrightarrow, "\zeta_{c, d_0}(f)*g"']&d_0\ar[d, "\zeta_{c,d_0}(f)"', twoheadrightarrow]\ar[dd, "f", bend left]\\
\bullet \ar[r, rightarrowtail]
% \ar[rd, "m_{fg}"', rightarrowtail, bend right]
& \bullet\ar[d, rightarrowtail, "m_f"']\\
&c
\end{tikzcd}
\]
\end{proof}
Then, we will see that $\{\zeta_c \colon \yo(c) \to \QuoE\}$ is a cocone. Notice that a family of natural transformations $\{\alpha_c\colon \yo(c) \to P\}_{c\in \ob (\C)}$ is a cocone under the functor $\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$ if and only if, for any composable maps
\[
\begin{tikzcd}
d \ar[r, rightarrow, "{f}"]& c_0 \ar[r, rightarrowtail ,"{m\in \MC}"]& c_1,
\end{tikzcd}
\]
where $m$ is an $\MC$-morphism, the equation
\begin{equation}\label{eq:coconecondition}
\alpha_{c_0,d}(f) = \alpha_{c_1,d}(mf).
\end{equation}
holds.
\[
\begin{tikzcd}
\C(d,c_0)\ar[rd, "\alpha_{c_0,d}"']\ar[rr, "m_{*}"]&&\C(d,c_1)\ar[ld, "\alpha_{c_1, d}"]\\
&P(d)&
\end{tikzcd}
\]
% For the later reference, we will provide a general
\begin{lemma}
The family of natural transformations $\{\zeta_c \colon \yo(c) \to \QuoE\}$ is a cocone under the functor $\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$.
\end{lemma}
\begin{proof}
For any composable maps
\[
\begin{tikzcd}
d \ar[r, rightarrow, "{f}"]& c_0 \ar[r, rightarrowtail ,"{m\in \MC}"]& c_1,
\end{tikzcd}
\]
where $m$ is an $\MC$-morphism, the equation \cref{eq:coconecondition}
\begin{equation*}
\zeta_{c_0,d}(f) = \zeta_{c_1,d}(mf).
\end{equation*}
holds, since post-composition with the $\MC$-morphism $m$
\[
\begin{tikzcd}
d \ar[rr, rightarrow, "{f}"]\ar[rd, "\zeta_{c_0, d}(f)"', twoheadrightarrow]&& c_0 \ar[r, rightarrowtail ,"{m\in \MC}"]& c_1\\
&\bullet\ar[ru, "m_f"', rightarrowtail]&&
\end{tikzcd}
\]
does not change the $\EC$-part of its factorization.
\end{proof}
\begin{lemma}
The family of natural transformations $\{\zeta_c \colon \yo(c) \to \QuoE\}$ is a colimit cocone of the functor $\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$.
\end{lemma}
\begin{proof}
We have proven that it is a cocone. So it remains to prove the universality. Let us take an arbitrary cocone $\{\alpha_c\colon \yo(c) \to P\}_{c\in \ob (\C)}$ under the functor. We will show the unique exsitence of the cocone map $\gamma \colon \QuoE \to P$.
First we will see the uniqueness. If there exists such $\gamma$, then we have
\[
\gamma_{c}(\pi \colon c \twoheadrightarrow q) = \gamma_c (\zeta_{q,c}(\pi \colon c \twoheadrightarrow q)) = \alpha_{q,c}(\pi \colon c \twoheadrightarrow q).
\]
since we have the commutative diagram
\[
\begin{tikzcd}
&\yo(q)\ar[ld, "\zeta_q"']\ar[rd, "\alpha_q"]&\\
\QuoE\ar[rr, "\gamma"']&&P
\end{tikzcd}
\]
and hence
\[
\begin{tikzcd}
&\C(c,q)\ar[ld, "\zeta_{q,c}"']\ar[rd, "\alpha_{q,c}"]&\\
\QuoE(c)\ar[rr, "\gamma_c"']&&P(c).
\end{tikzcd}
\]
This proves the uniqueness.
It suffices to prove that the function\footnote{Technically, we need to show the well-definedness, since this definition a priori depends on the choice of $\pi$. This well-definedness follows from the fact that the right class $\MC$ of a factorization system contains all isomorphisms.} $\gamma_{c}(\pi \colon c \twoheadrightarrow q)= \alpha_{q,c}(\pi \colon c \twoheadrightarrow q).
$ actually defines a cocone map.
The naturality with respect to $c\in \C$ follows since, for any
for
\[
\begin{tikzcd}
c_0\ar[r, "\forall f"]\ar[d, "\pi*f"', twoheadrightarrow]&c\ar[d, "\pi", twoheadrightarrow]\\
q_0\ar[r, rightarrowtail]&q,
\end{tikzcd}
\]
we have
\[
\gamma_{c}(\pi)*f= \alpha_{q,c}(\pi)*f = \alpha_{q, c_0}(\pi f) = \alpha_{q, c_0}(m_{\pi f} \circ (\pi*f)) = \alpha_{q_0,c_0}(\pi * f) = \gamma_{c_0}(\pi*f),
\]
where the fourth equation follows due to \cref{eq:coconecondition}.
\memo{The commutativity as a cocone map follows similarly by \cref{eq:coconecondition}}
\end{proof}
\printbibliography
\end{document}