A relational obstruction to transport through a quotient

Calculation in relations

Let $A=\{a\}$, $B=\{u,v\}$, and $C=\{c\}$, with $u\ne v$. Take

Then $S\circ R=\varnothing$.

If $q:B\to\{*\}$ identifies $u$ and $v$, the transported relations are

and $S'\circ R'=\{(a,c)\}$. Thus changing the middle object by a quotient can create new relational composites.

Scope

This is an elementary counterexample for arbitrary binary relations. It is not yet a counterexample for a specified class of legal total deterministic game strategies. Such a claim must show that $R,S$ represent allowed strategies, that $q$ is an allowed map, and that the stated transport agrees with strategy composition.

Note 13 discusses the legality issue. The double-category question remains open at that level.