Let $U: \textbf{Top} \to \textbf{Set}$ that forgets topology and $F: {\textbf{Set}} \to {\textbf{Top}}$ functor that maps set to discrete topology space over that set, namely $FX = (X, \mathcal D)$.
I want to formally prove these two are adjoint functors. To do that I want to prove that mapping $$\phi_{X,Y}:Hom_\textbf{Top}(FX,Y) \to Hom_\textbf{Set}(X,UY)$$ defined as $\phi_{X,Y} = U(g)$ is an isomorphism and natural in both arguments. I succeded in proving naturality using definition, but I am not sure if this is an isomorphism. I would like to somehow define $$\psi_{X,Y} : Hom_\textbf{Set}(X,UY) \to Hom_{\textbf{Top}}(FX,Y)$$ such that $$\psi_{X,Y} \circ \phi_{X,Y} = {1}_{Hom_{{\textbf{Top}}}(FX,Y)}$$ and $$\phi_{X,Y} \circ \psi_{X,Y} = 1_{Hom_{{\textbf{Set}}}(X,UY)},$$ but here I am stuck.
The sets $\text{Hom}_{\mathbf{Top}}(FX,Y)$ and $\text{Hom}_{\mathbf{Set}}(X,UY)$ are not only isomorphic, they are identical. A continuous map $FX \to Y$ is first and foremost a set map $X \to UY$, and any set map $X \to UY$ is automatically a continuous map $FX \to Y$. You have already defined $\phi_{X,Y}(g) = U(g)$. This is the identity map, which is a bijection.