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:

$$\gdef{\colim}{\underrightarrow{lim}\;} \gdef{\lim}{\underleftarrow{lim}\;} \gdef{\naturalto}{\Rightarrow}$$

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 $$\lim F = F(0_J)$$.