Let $\mathscr{M}$ be a model category and let $\mathscr{I}$ be a small category. Consider any homotopy colimit functor $\text{hcolim}_{\mathscr{M}}^{\mathscr{I}}\colon\mathscr{M}^{\mathscr{I}}\longrightarrow \mathscr{M}$ of shape $\mathscr{I}$ on $\mathscr{M}$. Here I adopt the notion of homotopy colimit considered in Homotopy Limit Functors on Model Categories and Homotopical Categories where a homotopy colimit is defined as a left approximation of the colimit functor.
Given a functor $X\colon\mathscr{I}\longrightarrow \mathscr{M}$, let me call a cocone $X\rightarrow P$ from $X$ (that is, a natural transformation $X\rightarrow cP$, where $P\in\mathscr{M}$ and $c$ is the constant functor $\mathscr{M}\longrightarrow \mathscr{M}^{\mathscr{I}}$) a homotopy colimit cocone if the canonical composite map $$ \text{hcolim}_{\mathscr{M}}^{\mathscr{I}}X\rightarrow \text{colim}_{\mathscr{M}}^{\mathscr{I}}X\rightarrow P $$ is a weak equivalence. Note that, given another homotopy colimit functor $\textbf{hcolim}_{\mathscr{M}}^{\mathscr{I}}$ of shape $\mathscr{I}$ over $\mathscr{M}$, the map $\text{hcolim}_{\mathscr{M}}^{\mathscr{I}}X\rightarrow P$ is a weak equivalence if and only if $\textbf{hcolim}_{\mathscr{M}}^{\mathscr{I}}X\rightarrow P$ is a weak equivalence because $\text{hcolim}_{\mathscr{M}}^{\mathscr{I}}$ and $\textbf{hcolim}_{\mathscr{M}}^{\mathscr{I}}$ are naturally weakly equivalent as functors over the colimit functor $\text{colim}_{\mathscr{M}}^{\mathscr{I}}$.
Let me now say that a functor $F\colon \mathscr{M}\longrightarrow \mathscr{N}$ between model categories preserves homotopy colimit cocones if, for every $X\in\mathscr{M}^{\mathscr{I}}$ (for a small category $\mathscr{I}$) and every homotopy colimit cocone $X\rightarrow P$ from $X$, $F\circ X\rightarrow FP$ is a homotopy colimit cocone from $F\circ X$.
Of course, we can dually define the concepts of homotopy limit cones and of preserving homotopy limit cones.
I would like to know if these notions are somehow well understood and/or useful. I am in particular interested in the following questions which I can not answer:
1) Let $F\colon \mathscr{M}\longrightarrow \mathscr{N}$ be a left Quillen functor and let $Q\colon\mathscr{M}\longrightarrow \mathscr{M}$ be a cofibrant replacement functor on $\mathscr{M}$. Does $F\circ Q$ preserve homotopy colimit cocones (and dually for a right Quillen functor)?
2) If $F\colon \mathscr{M}\longrightarrow \mathscr{N}$ is a left Quillen functor which is part of a Quillen equivalence and $Q$ is a cofibrant replacement functor on $\mathscr{M}$, does $F\circ Q$ preserve homotopy limit cones (and dually for a right Quillen functor)?
