Call a diagram $E$ in a model category a homotopy colimit diagram if the morphism $$\mathrm{hocolim}~E\to \mathrm{colim}~ E$$ is a weak equivalence. A homotopy colimit is defined as the categorical colimit of a cofibrant replacement of the diagram in the projective model structure and this is where the morphism comes from.
Let $F:C\rightleftarrows D:G$ be a Quillen equivalence between model categories $C$ and $D$. The (Edit: derived!) left adjoint $F$ preserves homotopy colimits, i.e. if $E$ is a homotopy colimit diagram in $C$, then $F\circ Q\circ E$ is a homotopy colimit diagram in $D$ where $Q$ denotes a cofibrant replacement.
Does the (Edit: derived!) right adjoint $G$ preserve homotopy colimits if the adjunction is a Quillen equivalence?
To be more precise, if $E$ is a homotopy colimit diagram in $D$, is $G\circ R\circ E$ is a homotopy colimit diagram in $C$ where $R$ denotes a fibrant replacement?
I suppose that this is true since the notion of homotopy colimit should depend only on the homotopy category and not on the model, I guess, but I cannot think of an argument.
Best Answer
The homotopy colimit functor $Ho(D^I)\rightarrow Ho(D)$ is the left adjoint of the constant diagram functor $Ho(D)\rightarrow Ho(D^I)$. Quillen equivalences induce Quillen equivalences between diagram categories, you you can replace $D$ with $C$, hence you're done by uniqueness of adjoints.
PS Don't worry about the fact that $D^I$ may not be a model category if $D$ is not cofibrantly generated. You can work with weaker axioms and convenient replacement of the notion of Quillen equivalence.