Categories and Sheaves Kashiwara Schapira Theorem 14.4.5

Question

In the proof of the theorem I have problems understanding why the functor $\operatorname{Hom}_\mathcal{C}$ has a right derived functor, as C is not necessarily a Grothendieck category. Moreover, but very likely directly related to that, I have problems understanding why in the last equation we have $R \operatorname{Hom}_\mathcal{C}(X,R\operatorname{F}(I))= \operatorname{Hom}^\bullet_\mathcal{C}(X,\operatorname{F}(I))$ for any $X\in \tilde{\mathcal{P}}$ and $I$ homotopically injective. It is likely that a solution to one of these two issues immediately solves the other one. Could anyone help?