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 \(J\):

commuting diagram

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

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