Suppose $F,G$ are adjoint functors with $\operatorname{Hom}(FX,Y)\cong \operatorname{Hom}(X,GY)$ where $F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{D}\to\mathcal{C}$ with $X\in\mathcal{C}$ and $Y\in\mathcal{D}$.
$\textbf{Q:}$ If $\phi:FX\to Y$ surjection or mono, can I conclude anything about mono or epi property of $\phi':X\to GY$ under identification of $\operatorname{Hom}(FX,Y)\cong \operatorname{Hom}(X,GY)$ by $\phi\to \phi'$? I have tried to see whether I can conclude mono or epi of $\phi'$. However, I can't conclude anything as I either have to modify domain or I can't test everything in the range part.
Let's take a classic adjoint: $G$ the forgetful functor from $\mathbf{Ab}$ to $\mathbf{Set}$, and $F$ the functor taking a set to the free Abelian group on it. Let $X$ be any nonempty set and $Y=FX$, the free group on $X$. If $\phi$ is the identity map on $FX$, it corresponds to the map $\phi':X\to FX$ taking each $x\in X$ to the corresponding generator of $FX$. This $\phi'$ is certainly not a surjection.