A functor $F : C \to D$ is final (often called cofinal), if we can restrict diagrams on $D$ to diagrams on $C$ along $F$ without changing their colimit.
Dually, a functor is initial (sometimes called co-cofinal) if pulling back diagrams along it does not change the limits of these diagrams.
Beware that this property is pretty much unrelated to that of a functor being an initial object or terminal object in the functor category $[C,D]$. The terminology comes instead from the fact that an object $d\in D$ is initial (resp. terminal) just when the corresponding functor $d:1\to D$ is initial (resp. final).
A functor $F : C \to D$ is final if for every object $d \in D$ the comma category $(d/F)$ is non-empty and connected.
A functor $F : C \to D$ is initial if the opposite $F^{op} : C^{op} \to D^{op}$ is final, i.e. if for every object $d \in D$ the comma category $(F/d)$ is non-empty and connected.
Let $F : C \to D$ be a functor
The following conditions are equivalent.
$F$ is final.
For all functors $G : D \to Set$ the natural function between colimits
is a bijection.
For all categories $E$ and all functors $G : D \to E$ the natural morphism between colimits
is a isomorphism.
For all functors $G : D^{op} \to Set$ the natural function between limits
is a bijection.
For all categories $E$ and all functors $G : D^{op} \to E$ the natural morphism
is an isomorphism.
For all $d \in D$
If $F : C \to D$ is final then $C$ is connected precisely if $D$ is.
If $F_1$ and $F_2$ are final, then so is their composite $F_1 \circ F_2$.
If $F_2$ and the composite $F_1 \circ F_2$ are final, then so is $F_1$.
If $F_1$ is a full and faithful functor and the composite is final, then both functors seperately are final.
The generalization of the notion of final functor from category theory to (∞,1)-higher category theory is described at
The characterization of final functors is also a special case of the characterization of exact squares.
If $D$ has a terminal object then the functor $F : {*} \to D$ that picks that terminal object is final: for every $d \in D$ the comma category $d/F$ is equivalent to $*$. The converse is also true: if a functor $*\to D$ is final, then its image is a terminal object.
In this case the statement about preservation of colimits states that the colimit over a category with a terminal object is the value of the diagram at that object. Which is also readily checked directly.
Every right adjoint functor is final.
Let $(L \dashv R) : C \to D$ be a pair of adjoint functors.To see that $R$ is final, we may for instance check that for all $d \in D$ the comma category $d / R$ is non-empty and connected:
It is non-empty because it contains the adjunction unit $(L(d), d \to R L (d))$. Similarly, for
two objects, they are connected by a zig-zag going through the unit, by the universal factorization property of adjunctions
The inclusion $\mathcal{C} \to \tilde \mathcal{C}$ of any category into its idempotent completion is final.
See at idempotent completion in the section on Finality.
The inclusion of the cospan diagram into its cocone
is initial.
By the characterization (here) of limits in a slice category, this implies that fiber products in a slice category are computed as fiber products in the underlying category, or in other words that dependent sum to the point preserves fiber products.
final functor, cofinal diagram
Section 2.5 of
Section 2.11 of
Notice that this says “final functor” for the version under which limits are invariant.
Section IX.3 of