Categories of modules and inverse homomorphisms $\DeclareMathOperator\Hom{Hom}$I have two equivalent rings $R$ and $S$ with $F$ and $G$ inverse equivalent functors, $F : {}_{R}\mathsf{M} \to {}_{S}\mathsf{M}$ and $G$ inverse. So I have 2 natural isomorphisms $\eta:GF \to 1_{{}_{R}\mathsf{M}}$ and $\zeta:FG \to 1_{{}_{S}\mathsf{M}}$.
This means that for every left $R$-module $M$ and for every left $S$-module $N$ there are isomorphisms
\begin{gather*}
  \eta_M:GF(M) \to M \\
  \zeta_N: FG(N) \to N
\end{gather*}
such that for every morphism $f:M \to M'$ and $g:N \to N'$ the diagrams are commutative, so:
\begin{align*}
  \eta_{M'} \circ GF(f) &= f \circ \eta_M \\
  \zeta_{N'} \circ FG(g) &= g \circ \zeta_N
\end{align*}
For every left $R$-module $M$ and every left $S$-module $N$ I have two isomorphisms of abelian groups:
\begin{align*}
  \phi:\Hom_S(N,F(M)) &\to \Hom_R(G(N), M) \\
  \theta:\Hom_S(F(M),N) &\to \Hom_R(M,G(N))
\end{align*}
such that
\begin{align*}
  \phi&: \gamma \to \eta_M\circ G(\gamma) \\
  \theta&: \delta \to G(\delta) \circ \eta_{M}^{-1}
\end{align*}
I need to write down what $\phi^{-1}$ and $\theta^{-1}$ are. Can someone help me? Thanks!
 A: $\DeclareMathOperator\Hom{Hom}$Let $\gamma:N\to F(M)$ and $\phi(\gamma)=\eta_M\circ G(\gamma):G(N)\to M$.
Then $F(\eta_M^{-1}\circ\phi(\gamma))=FG(\gamma)=\zeta_{F(M)}^{-1}\circ\gamma\circ\zeta_N$ by the commutativity of the following diagram:
$$
  \require{AMScd}
  \begin{CD}
    FG(N)
    @> \zeta_N >>
    N
    \\
    @V FG(\gamma) VV
    {}
    @VV \gamma V
    \\
    FGF(M)
    @>> \zeta_{F(M)} > F(M)
  \end{CD}
$$
Consequently $\gamma=\zeta_{F(M)}\circ F(\eta_M^{-1}\circ\phi(\gamma))\circ\zeta_N^{-1}$
hence
$$\phi^{-1}(\xi)=\zeta_{F(M)}\circ F(\eta_M^{-1}\circ\xi)\circ\zeta_N^{-1}.$$

$\newcommand\Mod[1]{{}_{#1}\mathsf{M}}\newcommand\op{\mathsf{op}}\newcommand\Ab{\mathsf{Ab}}$We have two functors
\begin{align}
&\Hom_S(-,F(-)):\Mod S^\op\times\Mod R\to\Ab,&
&\Hom_R(G(-),-):\Mod S^\op\times\Mod R\to\Ab
\end{align}
controvariant in the first entry and covariant in the second entry.
For an $S$-module $N$ and an $R$-module $M$, the sets $\Hom_S(N,F(M))$ and $\Hom_R(G(N),M)$ are abelian groups respect to pointwise addition.
For example, if $\gamma_1,\gamma_2\in\Hom_S(N,F(M))$ then
\begin{align}
&\gamma_1+\gamma_2\in\Hom_S(N,F(M)),&
&(\gamma_1+\gamma_2)(x)=\gamma_1(x)+\gamma_2(x),&
&(x\in N).
\end{align}
Then we have a natural isomorphism $\phi:\Hom_S(-,F(-))\to\Hom_R(G(-),-)$.
