2 fixed typo

I'm in need of a condition that is analogous to the "finality" condition in the following lemma:

Lemma: A functor $F\colon A\to B$ is final if and only if for any functor $x\colon B\to Set$, the natural map $colim (xF)\to colim(x)$ is an isomorphism.

This lemma could be taken instead as a definition of final functor, but finality is more easily recognized by whether all slice categories of a certain kind are non-empty and connected. I want a recognition principle for a more general kind of finality, which I'm calling local finality.

The more general context requires a bit of notation. If $A$ is a category, write $A-Set$ for the category of functors $A\to Set$. If $F\colon A\to B$ is a functor, write $\Delta_F\colon B-Set\to A-Set$ for the composition with $F$" functor, and write $\Sigma_F$ for its left adjoint and $\Pi_F$ for its right adjoint (these three are also sometimes denoted by $F^*, F_!$, and $F_*$ respectively).

The following lemma (obviously) holds for some appropriate definition of locally final.

Lemma: Suppose that we have a commutative diagram $A\xrightarrow{F}B\xrightarrow{x}C$ and let $G:=xF$. Then $F$ is locally final if and only if the natural map $\Sigma_G\Delta_F\to\Sigma_F$ \Sigma_G\Delta_F\to\Sigma_x$is an isomorphism. Is there a nice recognition principle for this kind of local finality"? I have a big messy condition obtained by following my nose, but it's of no use. Any help would be greatly appreciated. Thanks! 1 # Local finality condition (for re-indexing parameterized colimits) I'm in need of a condition that is analogous to the "finality" condition in the following lemma: Lemma: A functor$F\colon A\to B$is final if and only if for any functor$x\colon B\to Set$, the natural map$colim (xF)\to colim(x)$is an isomorphism. This lemma could be taken instead as a definition of final functor, but finality is more easily recognized by whether all slice categories of a certain kind are non-empty and connected. I want a recognition principle for a more general kind of finality, which I'm calling local finality. The more general context requires a bit of notation. If$A$is a category, write$A-Set$for the category of functors$A\to Set$. If$F\colon A\to B$is a functor, write$\Delta_F\colon B-Set\to A-Set$for the composition with$F$" functor, and write$\Sigma_F$for its left adjoint and$\Pi_F$for its right adjoint (these three are also sometimes denoted by$F^*, F_!$, and$F_*$respectively). The following lemma (obviously) holds for some appropriate definition of locally final. Lemma: Suppose that we have a commutative diagram$A\xrightarrow{F}B\xrightarrow{x}C$and let$G:=xF$. Then$F$is locally final if and only if the natural map$\Sigma_G\Delta_F\to\Sigma_F\$ is an isomorphism.

Is there a nice recognition principle for this kind of local finality"? I have a big messy condition obtained by following my nose, but it's of no use. Any help would be greatly appreciated.

Thanks!