A right Kan extension approach to finite presheaves

Aim

Find a criterion for $[\mathcal C^{\mathrm{op}},\mathbf{FinSet}]$ to be an elementary topos or locally cartesian closed, as asked in Finite presheaf topoi.

Proposed approach

Use a left-exact comonad to reduce the question to existence of suitable right Kan extensions. The motivating idea is to understand which indexing categories have limits for every finite-set-valued diagram, then impose corresponding conditions on comma categories.

This is an approach to formulate, not an established reduction. A particular comonad and adjunction have not yet been specified. In particular, no right adjoint to the inclusion of finite-valued presheaves into all presheaves is assumed here.

Next step

Specify the categories, functors, and adjunction involved. Write the proposed right Kan extension pointwise as a limit, then prove both its finiteness and the implication for exponentials or the subobject classifier. Compare the resulting condition with the existing sufficient condition that every slice $\mathcal C/c$ is essentially finite.

The underlying finite-presheaf question was discussed with Jérémie Marquès.