As is described for example in Cisinski's book on Higher Category theory, the left (right) derived functor of a functor between model categories $F: \mathcal{C} \rightarrow \mathcal{D} $ can be described as the right (left) Kan extension of $F$ along the locolization functor $\gamma_\mathcal{C}: \mathcal{C} \rightarrow \operatorname{ho}(\mathcal{C})$, ie. $\mathbb{L}F=\gamma_\mathcal{D}\circ \operatorname{Ran}_{\gamma_\mathcal{C}}F: \operatorname{ho}(\mathcal{C}) \rightarrow \operatorname{ho}(\mathcal{D})$. Obviously, this does not depend on the cofibrations and fibrations in $\mathcal{C}$, since the localization happens only with respect to the weak equivalences. So (although a lot of structure might be lost) this definition does also make sence for general Categories with weak equivalences.
I will refer to the definition of the left/right satellite functor as it is given in Weibel, for $F$ a functor between abelian categories that are allowed explicitly to not have enough projectives or injectives, it is a universal $\delta$-functor with $F$ as its zero-component. By universality, it is essentially unique, and if both exist equal to the derived functor, although it does exist in the cases specified above where a (classical) derived functor does not exist.
In the case where there are not enough projectives and injectives, there is no projective/ injective model structure on the category of chain complexes on our abelian category, but one still give it the structure of a category with weak equivalences using the quasi-isomorphisms. My question is, as both definitions make sense in this context, the one via Kan extensions and the one via universal $\delta$-functors, whether they are equal (as in the usual context) or not, and which one should be considered the "right" definition of a derived functor in this case. Greetings,
Intergalakti