Given any category $C$, let $\mathrm{Epi}(C)$ and $\mathrm{Mono}(C)$ denote the (generally non-full) subcategories of $C$ consisting of the epimorphisms and monomorphisms respectively.
Then, is any monomorphism in $\mathrm{Epi}(C)$ also a monomorphism in $C$ (or dually, is any epimorphism in $\mathrm{Mono}(C)$ also an epimorphism in $C$)?
If $C$ has (co)kernel pairs, then the answer is positive. Indeed, if $C$ has kernel pairs and $f$ is a monomorphism in $\mathrm{Epi}(C)$, then for the two maps $p$ and $q$ in the kernel pair of $f$ in $C$, one has that $p$ and $q$ are epimorphisms (split by the diagonal), and so $f \circ p=f \circ q$ would imply that $p=q$ (as $f$ is monic in $\mathrm{Epi}(C)$), and so $f$ must in fact be monic in $C$. Dually, if $C$ has cokernel pairs, then any epimorphism in $\mathrm{Mono}(C)$ is also an epimorphism in $C$.
In the general case, I am not sure if the answer is positive or negative.
The answer should be negative.
Consider the category which has three distinct objects $0,1,2$ and five distinct nontrivial arrows: $\alpha,\beta:0\to1$, $\gamma,\delta:1\to2$ and $\epsilon:0\to2$. Composition is defined in the only way possible, with $\gamma\alpha$ and all other such similar compositions set equal to $\epsilon$.
Evidently $\alpha,\beta$ are not epimorphisms and $\gamma,\delta$ are not monomorphisms. However, for trivial reasons $\gamma,\delta$ are epimorphisms (and $\alpha,\beta$ are monomorphisms). Within the subcategory of objects and epimorphisms, $\gamma,\delta$ are the only nontrivial arrows and they are easily monomorphisms in this category. But, stepping outside into the real world, we see they are no longer monomorphisms. (and $\alpha,\beta$ are epimorphisms in the category of monomorphisms but are no longer epimorphisms in the original category).