I’m currently working through Mac Lane’s and Moerdijk’s “Sheaves in Geometry and Logic”, and came upon the following two sentences in a proof (p. 43):

Note that when $P = Hom(-, C) = \mathbf{y}C$ is representable the corresponding category of elements $\int P$ has a terminal object—the element $1 : C \to C$ of $P(C)$. Therefore the colimit of the composite $A \circ \pi_P$ will just be the value of $A \circ \pi_P$ on the terminal object.

It was not immediately obvious to me why the colimit of a functor should be the image of the terminal object, though I got the intuition that we automatically had a cocone. Sheldon Axler gives the following advice to students in the preface of his “Linear Algebra Done Right”:

You cannot expect to read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast.

Taking this advice to heart, I set out to prove: $\DeclareMathOperator{\colim}{\underset{\longrightarrow}{lim}}\DeclareMathOperator{\lim}{\underset{\longleftarrow}{lim}}\newcommand{\naturalto}{\overset{\bullet}{\longrightarrow}}$

Proposition. Let $F : J \to C$ be a functor and $J$ a diagram with terminal object $1_J$. Then the colimit $\colim F$ exists and $\colim F = F(1_J)$.

In other words, if a diagram has a terminal object, then the colimit is trivial: the colimit is the image of the terminal object.

Proof. Let $\tau : F \naturalto \Delta(F(1_J))$ be the natural transformation given by $\tau_j = F(t_j) : F(j) \to F(1_J)$ where $t_j : j \to 1_J$ is the unique morphism in $J$ given by the terminality of $1_J$. Then this clearly defines a cocone from $F$ to $F(1_J)$.

Let $\phi : F \naturalto \Delta(N)$ be any other cocone on $F$. Then there exists a morphism from $F(1_J)$ to $N$, namely, $\phi_{1_J}$. Moreover, we claim this morphism makes the following diagram commute for all $f : j \to k$ in $J$:

Indeed, since the $\tau_i$ are $F(t_i)$, this is immediate from the naturality of $\phi$. We claim $\phi_1 : F(1_J)$ is the unique such map. Let $\rho : F(1_J) \to N$ be any other morphism making the diagram commute. Then in particular, the diagram

commutes. But $\tau_1 = F(t_1 : 1_J \to 1_J) = F(1_{1_J} : 1_J \to 1_J) = 1_{F(1_J)}$. So $\phi_{1_J} = \rho \circ 1_{F(1_J)} = \rho$.

We thus conclude that the colimit exists and is exactly $F(1_J)$. Q.E.D.

Corollary. Let $F : J \to C$ be a functor and $J$ a diagram with initial object $0_J$. Then the limit $\lim F$ exists and $\lim F = F(0_J)$.