I have a map $\require{AMScd}f\colon X \to Y$ in some category $\mathcal E$ which I would like to show is epic. However the only description I have of $X$, $Y$, and $f$ is through the Yoneda embedding, in that I have an explicit description only of the functors $X_\ast,Y_\ast \colon \mathcal C^{\mathrm{op}} \to \mathbf{Set}$ and of the natural transformation $f_\ast \colon X_\ast \to Y_\ast$. (I am writing $Z_\ast = \hom({-},Z)$.)
Had I to show $f$ monic I would be happy since that can be formulated through limits, which the Yoneda embedding preserves. Unfortunately I am sad as it does not preserve colimits.
In my particular case $\mathcal E$ is a topos; I have been shown one way in this context: $f$ is epic iff every generalized element $q\colon V \to Y$ is covered via $f$ by an element $p\colon U \to X$, in that there is an epi $u\colon U \twoheadrightarrow V$ such that the square commutes: \begin{CD} U @>u>> V \\\\ @VpVV @VVqV \\\\ X @>>f> Y . \end{CD} Through Yoneda this translates to: For every $q \in Y_\ast V$ there are $p \in X_\ast U$ and $u\colon U \twoheadrightarrow V$ such that $f_\ast(p) = qu \in Y_\ast U$.
This is good, but I have no concrete description in the category $\mathcal E$ of the arrow $q\colon V\to Y$, so it is hard to determine its composite with $u\colon U\to V$ which must be epic in $\mathcal E$.
Are there any other ways to prove $f$ epic using $f_\ast$?
First of all, a small correction: note that if $X$ is an object of $\mathcal{C}$, then $\operatorname{Hom}({-}, X)$ is actually a contravariant functor $\mathcal{C}^{\operatorname{op}} \to \mathbf{Set}$.
Now, recall that this functor $\operatorname{Hom}({-}, X)$ acts on morphisms $\varphi \in \operatorname{Hom}(U, V)$ precisely by sending $x \in \operatorname{Hom}(V, X) \mapsto x \circ \varphi \in \operatorname{Hom}(U, X)$. Therefore, in terms of the functors $X_*$ and $Y_*$, the condition you want is precisely that for every object $U$ and element $y \in Y_*(U)$, there exists an epic morphism $\varphi \in \operatorname{Hom}(V, U)$ and an element $x \in X_*(V)$ such that $$Y_*(\varphi)(y) = f_*(x).$$