As far as I understand it, for $R,S$ rings and $M$ an $R$-module ,$N$ and $R,S$-bimodule and $L$ an $S$-module, we have an isomorphism
$$ \text{RHom}_R(N \otimes_S^L L, M) \cong \text{RHom}_S(L,\text{RHom}_R(N,M)). $$
To what extent is there a similar isomorphism of $R\mathcal H\text{om}$ for, say, ringed spaces $(X,O_X)\xrightarrow f(Y,O_Y)$ and modules of $O_Y$- and $O_X$-sheaves? Or maybe for simplicity set $Y=X$, with a map $O_Y \rightarrow O_X$. Feel free to adjust the assumptions as necessary.
I tried searching the literature, but couldn't find a satisfying answer. The closest I could find was https://stacks.math.columbia.edu/tag/0DVC.
A reference would be great!
See Kashiwara, Schapira, Categories and Sheaves, Theorem 18.6.9 (iii) eqn (18.6.13): Let $f : X \to Y$ be a morphism of ringed sites. There are isomorphisms \begin{align} Rf_{\ast}R\mathcal{Hom}_{\mathcal{O}_{X}}(Lf^{\ast}G,F) \simeq R\mathcal{Hom}_{\mathcal{O}_{Y}}(G,Rf_{\ast}F) \end{align} functorial with respect to $F \in D(\mathcal{O}_{X})$ and $G \in D(\mathcal{O}_{Y})$.
In the Stacks Project, I could only find the case when $f$ is the identity morphism $\mathrm{id}_{X}$, see Tag 08DJ.