= Solution
For $A\in\mathcal C$ and $F:\mathcal C\to\mathbf{Set}$, the <Yoneda lemma> gives a natural bijection
$$
\operatorname{Nat}(\mathcal C(A,-),F)\cong F(A),
\qquad \alpha\longmapsto\alpha_A(1_A).
$$
Assume $\mathcal C$ 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 is
$$
\coprod_{A\in\mathcal C}\coprod_{x\in F(A)}
\mathcal C(A,-)\longrightarrow F.
$$
At 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>.
Back to article page
