If the category $J$ has an initial object $s$, prove that every functor $F: J \rightarrow C$ to any category $C$ has a limit, namely $F(s)$.

Question

Exercise 3.4.3 in MacLane's Categories for the Working Mathematician.

I am able to show that $F(s)$ is a cone, but unable to show that is universal among other cones with base $F$. How do I establish the unique arrow from any other vertex to $F(s)$? It feels as though I am missing something obvious. Thanks!

Answer 1

A cone $x \to F$ consists of compatible maps $x \to F(j)$, in particular we get $x \to F(s)$. Since for arbitrary $j$ there is a unique morphism $s \to j$, we see that $x \to F(j)$ factors as $x \to F(s) \to F(j)$. Hence, a cone is just a morphism $x \to F(s)$. This shows that $\lim F = F(s)$.