Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 1 / a / Solution

For $A\in \mathcal{C}$ and $F:\mathcal{C}\to \mathbf{Set}$, the Yoneda lemma gives a natural bijectionAssume $\mathcal{C}$ is small. For every pair $(A,x)$ with $x\in F(A)$, let ${\alpha }^{A,x}:\mathcal{C}(A,-)\to F$ be the corresponding natural transformation. Their copairing isAt an object $B$, the element $x\in F(B)$ is the image of $1_B$ in the summand indexed by $(B,x)$. The map is therefore pointwise surjective and hence an epimorphism in the functor category.