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!

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 $\Sigma_G\Delta_F\to\Sigma_x$ 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!