Let $F: \mathcal{C} \to \mathcal{D}$ be a left Quillen functor between model categories. In Definition 2.16 of Goerss-SchemmerhornGoerss–Schemmerhorn - Model Categories and Simplicial Methods, the left derived functor $LF: \mathcal{C} \to \operatorname{Ho}(\mathcal{D})$ is defined by $X \mapsto F(P)$. However, the article does not define the functor on maps.
In practice, this is often done under the assumption that $\mathcal{C}$ has functorial factorisation (or at least functorial cofibrant replacement), but Goerss-SchemmerhornGoerss–Schemmerhorn seem to imply that this assumption is not needed. Could someone help me understand a construction that works in general, without requiring functorial factorisation?
My best guess is to fix an arbitrary cofibrant replacement $P_X \to X$ for each $X \in \mathcal{C}$, and then try to construct a well-defined map $F(P_X) \to F(P_Y)$ in the homotopy category of $\mathcal{D}$, given a morphism $f:X\to Y$. We get morphisms $$ F(P_X) \rightarrow F(X) \overset{f}{\to} F(Y) \leftarrow F(P_Y). $$ If the right-hand arrow were a weak equivalence, then we would be done, since it would be invertible in the homotopy category. This is the case if $Y$ is cofibrant, since left Quillen functors preserve weak equivalences between cofibrant objects. They do not preserve weak equivalences in general though, so the approach doesn't seem to work.
