Final functor

Remarks

Here a category is connected when any two objects are joined by a finite zigzag of morphisms. If $F$ is final, then for every functor $G \colon \mathcal{D} \to \mathcal{E}$ the colimit of $G \circ F$ exists if and only if the colimit of $G$ exists. The canonical morphism $\operatorname{colim} (G \circ F) \to \operatorname{colim} G$ is then an isomorphism. For small $\mathcal{C}$ and $\mathcal{D}$ this property for all $G$ is equivalent to finality.

Example: let $\mathcal{D}$ have a terminal object $t$, and let $F$ be the functor from the one-object category picking out $t$. For each $d$ the category $d \downarrow F$ has one object, the unique arrow $d \to t$, and only its identity, so it is nonempty and connected. Hence $\operatorname{colim} G = G(t)$ for every $G$.

Non-example: let $\mathcal{D}$ be the poset $\set{0 < 1}$ and let $F$ pick out $0$. There is no arrow $1 \to 0$, so $1 \downarrow F$ is empty and $F$ is not final.

The inclusion of a cofinal subset of a directed set is a final functor.