Let $f: A \rightarrow B$ be an epic morphism in category $C$. Then is it true that the morphism $h: \text{Hom}(B,-)\rightarrow \text{Hom}(A,-)$ is mono? If yes why?
What if $B$ is the initial object? Then can I say because $\text{Hom}(B,-)$ is a singletone, then, $h$ is mono?
Yes. We wish to show that, for any functor $F:C\to\textbf{Set}$ and any two morphisms (ie, natural transformations) $\alpha,\beta:F\to\operatorname{Hom}(B,-)$, if we have $h\circ\alpha=h\circ\beta$, then we have $\alpha=\beta$. Now, showing $\alpha=\beta$ amounts to showing that $$\alpha_c=\beta_c:F(c)\to\operatorname{Hom}(B,c)$$ for every $c\in \operatorname{ob}(C)$. We know $h_c\circ\alpha_c=(h\circ\alpha)_c=(h\circ\beta)_c=h_c\circ\beta_c$ by hypothesis; in other words, we have $$\alpha_c(x)\circ f=(h_c\circ\alpha_c)(x)=(h_c\circ\beta_c)(x)=\beta_c(x)\circ f$$ for every $x\in F(c)$. Since $f$ is epic, this means $\alpha_c(x)=\beta_c(x)$ for every $x\in F(c)$, which means in particular that $\alpha_c=\beta_c$, as desired.