I am trying to understand how to prove properties involving derived functors which come from the property of the associated functors. I'll make an example to explain what I am saying: we know there's an adjunction $$ Hom_R( M \otimes_R N, H) \cong Hom_R ( M, Hom_R(N,H))$$
for $M,N,H \in Mod(R)$ which gives an isomorphism of functors (let us consider $H$ to be fixed). Then how can I prove: $$RHom_R(M^{\bullet} \otimes^L N^{\bullet}, H^{\bullet}) \cong RHom_R(M^{\bullet}, RHom_R(N^{\bullet}, H^{\bullet}))$$
for $M^{\bullet}, N^{\bullet} \in D^{-}(Mod(R)), H^{\bullet} \in D^{+}(Mod(R)$? I don't undestand whether I should look for a 'natural' isomorphism or do it by hand.