Given a model category $C$, a simplicial symmetric monoidal model category $D$ (in the sense of Goerss-Jardine) and a left Quillen functor $F:C\to D$, define $|F|:C\to\mathsf{sSet}$ to be the functor
$$|F|=\mathsf{Map}_D(I,F(-))$$
where $I$ is the unit of the monoidal structure of $D$ which we suppose to be cofibrant.
Then, is there a link between $|F|$ and $|\mathbb LF|$, like $\mathbb L|F|\simeq |\mathbb L F|$ ? The problem is that $|F|$ may not verify the classical conditions to have a left derived functor, as we want fibrations on the right argument of the mapping space.
In my case $D=C(k)$ for $k$ a field of characteristic $0$, so every object is fibrant, which might help. My functor $F$ computes something which has the good homotopy type only for cofibrant objects, so I'd want $\mathsf{Map}_D(I,F(Q(-)))$ to yield a functor $\mathsf{Ho}(C)\to\mathsf{Ho}({\mathsf{sSet}})$ and I'd like it to be the total left derived functor of $|F|$.
Best Answer
If I is cofibrant and all objects are fibrant, then Map(I,−) is automatically derived. Thus, precomposing with the cofibrant replacement functor Q suffices to derive both |F| and F.
We get L|F|=|QF| and |LF|=|QF|, so indeed there is a weak equivalence between L|F| and |LF|.