Let $f \colon X \to Y$ be a continuous map of locally compact spaces. Denote by $Sh(X)$, $Sh(Y)$ the categories of sheaves of $k$-vector spaces for some field $k$ and by $D^b(X)$, $D^b(Y)$ their bounded derived categories. Suppose that $f_!$ has a finite cohomological dimension so that the right adjoint $f^! \colon D^b(Y) \to D^b (X)$ to $\mathbf Rf_!$ exists. I want to show that there is a natural isomorphism $$ \mathbf Rf_!(f^{-1} F \otimes G) \cong F \otimes \mathbf Rf_! G, \quad F \in D^b(Y), \; G \in D^b(X). $$ This isomorphism was used, e.g. in the proof of Theorem 7 (Verdier duality), p. 11 of these notes.
I was trying to take an arbitrary $H \in D^b(Y)$ and write $$ \mathrm{Hom}_{D^b(Y)} (\mathbf Rf_!(f^{-1} F \otimes G),H) \cong \mathrm{Hom}_{D^b(X)}(f^{-1}F \otimes G,f^! H) \\ \cong \mathrm{Hom}_{D^b(X)} (G, \mathbf R\mathcal{Hom}(f^{-1} F,f^! H)) $$ Now, if I could use that $\mathbf R\mathcal{Hom}(f^{-1} F, f^! H) \cong f^! \mathbf R \mathcal{Hom}(F,H)$ then I could continue the above sequence of isomorphisms as $$ \cong \mathrm{Hom}_{D^b(X)} (G, f^! \mathbf R\mathcal{Hom}(F,H)) \cong \mathrm{Hom}_{D^b(Y)} (\mathbf Rf_! G, \mathbf R\mathcal{Hom}(F,H)) \\ \cong \mathrm{Hom}_{D^b(Y)}(F \otimes \mathbf Rf_! G, H). $$ By Yoneda this implies the required statement. But now I have to prove that $\mathbf R\mathcal{Hom}(f^{-1} F, f^! H) \cong f^! \mathbf R \mathcal{Hom}(F,H)$. I'm not sure that it is easier than the original statement. Please, help me.