Epimorphism in the functor category $[\mathbf{C}^{op}, \mathbf{Set}]$

Let $\mathcal{A}$ and $\mathcal{B}$ be categories, $\mathcal{F},\mathcal{G}\colon\mathcal{A}\to\mathcal{B}$ be functors, $\alpha\colon\mathcal{F}\to\mathcal{G}$ be a natural transformation.

If $\alpha(a)$ is an epimorphism for every $a\in\text{Obj}(A)$, then $\alpha$ is an epimorphism in $\text{Func}(\mathcal{A},\mathcal{B})$. It is an easy exercise.

If $\alpha$ is an epimorphism in $\text{Func}(\mathcal{A},\mathcal{B})$ and $\mathcal{B}$ is finitely cocomplete, then $\alpha(a)$ is an epimorphism in $\mathcal{B}$ for every $a\in\text{Obj}(\mathcal{A})$. For every object $a\in\text{Obj}(\mathcal{A})$ denote by $\Delta_a\colon\mathbf{1}\to\mathcal{A}$ such functor, that $\Delta_a(0)=a$. Note, that if $\mathcal{B}$ is finitely cocomplete, then the inverse image functor $\mathcal{B}^{\Delta_a}\colon\mathcal{B}^{\mathcal{A}}\to\mathcal{B}^{\mathbf{1}}$ is right exact. Therefore, the evaluation functor $\text{ev}_a\colon\mathcal{B}^{\mathcal{A}}\to\mathcal{B}$, such that $\text{ev}_a(\mathcal{T})=\mathcal{T}(a)$ for every $\mathcal{T}\in\text{Func}(\mathcal{A},\mathcal{B})$, which is isomorphic to $\mathcal{B}^{\Delta_a}$, preserves epimorphisms.

Then it is sufficient to note that $\mathbf{Set}$ is finitely cocomplete.

Of course, the only difficult part of the proof is the statement that the inverse image functor preserves pointwise colimits. You can read about the theory of pointwise limits/colimits in the Mac Lane's CFWM and in the Borceux's handbook.