This is a corrected follow up to When do finite sets embed in a category?
Let $C$ be an (finite) extensive category with terminal object $1$. Let $I$ be an index category. Let $j: \mathrm{FinSet}\to C$ be the functor induced by $*\mapsto 1$. Let $f$ be a finite sets-valued presheaf on $I$, that is a functor $I^{op}\to \mathrm{FinSet}$. Let $F$ be the associated category of elements.
Then my question is whether there is an isomorphism of categories? $$ [F, C]=[I^{op}, C]/j(f). $$