I have just recently started approaching the topic of derived categories in algebraic geometry, and I'm doing so reading Huybrechts "Fourier-Mukai transforms in algebraic geometry". I have a doubt when it comes to generalizing classical natural isomorphisms to the corresponding derived version.
Take for instance the projection formula. In its derived version, if $f:X\to Y$ is a proper morphism of projective schemes, it looks like $$ Rf_*(\mathcal{F}^\bullet)\otimes^L\mathcal{E}^\bullet \simeq Rf_*(\mathcal{F}^\bullet\otimes^L Lf^*\mathcal{E}^\bullet ) $$ for any two objects $\mathcal{F}^\bullet, \mathcal{E}^\bullet $ of the derived category of $X$, respectively $Y$.
We also have a classical version of this formula: this time $\mathcal{E}$ is locally free on $Y$ and $\mathcal{F}$ is a sheaf on $X$, the formula then is
$$ f_*(\mathcal{F}\otimes f^*\mathcal{E})\simeq f_*(\mathcal{F})\otimes \mathcal{E}. $$
My question is: how do I prove the derived one starting from the classical result? Is there anything to prove at all? Or is it just that, by the setup of derived functors, the latter yields immediately the first one? If this is the case, what am I missing of the theory of derived functors (i.e. how do you prove it?)
I'm wondering the same for other isomorphisms, like the canonical isomorphisms related to tensor products for instance..