Suppose

$$ mathsf{X}:mathsf{J} longrightarrow mathsf{CAT} $$

factors through the functor

$$ mathsf{Cat}^{mathsf{op}} longrightarrow mathsf{CAT} $$

which sends a small category $mathsf{A}$ to the category of presheaves $widehat{mathsf{A}}$ and sends a functor $f:mathsf{A} rightarrow mathsf{B}$ to the inverse image functor $f^*:widehat{mathsf{B}} rightarrow widehat{mathsf{A}}$. What further hypotheses are necessary so that the conical limit, in $mathsf{CAT}$ of the diagram $mathsf{X}$ may be computed as presheaves on the colimit of the factoring of $mathsf{X}$ through $mathsf{Cat}^{mathsf{op}}$?

I know from http://tac.mta.ca/tac/reprints/articles/25/tr25.pdf that this works for some limits taken in the category of toposes, but the forgetful functor$mathsf{Topos} longrightarrow mathsf{CAT}$ admits a right 2-adjoint, so in general those limits of toposes do not agree with the limits of the underlying categories.

However, this does work sometimes, for example:

-) products (definitely) (take the coproduct before taking presheaves)

-) op-lax limits of arrows (I think) (take the collage before taking presheaves)

but I’m really not clear on what’s known about this question which seems like the sort of thing which is “well known”.