Choosing projective replacement to be functorial

Question

A basic result of homological algebra says that if $\mathsf A$ is an abelian category with enough projectives, then the mapping $P:\mathsf{Obj}(\mathsf A)\rightarrow \mathsf{Obj}(\mathsf{K} ^+(\mathsf A))$ sending each object to the homotopy type of its projective resolutions almost lifts to a functor in the sense that $P(g\circ f)\cong P(g)\circ P(f)$.

1. Is there a general way to fix the "almost" functoriality of $P$? If not, are classical derived not actually functors?

In Riehl's Categorical Homotopy Theory, sec 2.3, the author says that in $R$-$\mathsf Mod$, a more refined process enables the replacement of any complex with a quasi-isomorphic complex of projectives. She then says that careful choice of projective resolution yields a proper functor $\mathsf{Ch}_\bullet ^+(R)\rightarrow \mathsf{Ch}_\bullet ^+(R)$ along with a natural quasi-isomorphism $q:Q\Rightarrow 1$. In other words, projective replacement is a left deformation of the homotopical category $\mathsf{Ch}_\bullet ^+(R)$ with q.i's as weak equivalences.

2. What is this "refined process" and "careful choice"?
3. What is a general condition on an abelian category with enough projectives that would ensure projective replacement become functorial and moreover a left deformation as in $R$-$\mathsf{Mod}$?

In Remark 2.3.2, the author notes the absense of functorial deformations, the usual process of projective replacement defines total derived functors, but not point-set level derived functors.

4. What is meant by functorial deformations?

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Update: This question seems to be related, but I don't see any conditions guaranteeing a "projective resolvent functor".

Answer 1

1. Derived functors are functors. What is your confusion? You divide out the chain homotopy relation in the target category.

2,3. In order to construct a functorial projective resolution you need to have for each object A in the abelian category a natural surjection from a projective object $P_A \to A$.

4. This is similar to functorial cofibrant replacements.

In most practical situations the functoriality exists. I'll give you one example where it does not: consider the abelian category of finitely generated abelian groups, then there enough projectives, but there is no way to choose them functorially.