I have been thinking about the following question: given a morphism of coloured dg-operads $$\phi: P \longrightarrow Q$$ we derive a lax morphism between their respective monads $T= T_P \longrightarrow T_Q =S$ where $T$ is a monad on a category $C$ and $S$ on a category $D$. As a result we have a base change functor between their categories of algebras $$\phi_*: S-Alg \longrightarrow T-Alg$$ In case $S-Alg$ has enough coequalizers, this functor has a left adjoint $\phi^{!}$ constructed using an induced colax morphism of monads $(f,\alpha)$, that is, a functor $f: C\longrightarrow D$ and a natural transformation $\alpha: fT \Longrightarrow Sf$, as follows: for $(X,\lambda) \in Alg(T)$, we obtain $\phi^{!}X$ as the coequalizer of $(Sf\lambda, \mu_{fX} \circ S\alpha_X)$, i.e.
Both categories $C$ and $D$ have a notion of quasi-isomorphisms (and model category structures) by considering them point-wise.
The functor $\phi^{!}$ acts on morphisms $r:X \longrightarrow Y$ of $T$-algebras as follows: it is the induced arrow in the diagram below
My question is the following: if the functors $S,T$ and $f$ preserve quasi-isomorphisms, does the functor $\phi^{!}$ preserve quasi-isomorphisms?
My first idea was to use homological algebra. However, the only result that I know would help requires that the map $\mu_{fX}\circ S\alpha_X - Sf \lambda_X$ is point-wise a monomorphism, i.e. then we could bottom and top row of the diagram to short exact sequences. However, this will not be the case.
My second idea would be to look at model category techniques. The objects $SfX$ and $SfTX$ are cofibrant as $S$-algebras, the two right vertical morphisms are weak equivalences and the two epimorphisms are fibrations. I do not know how to proceed next.
Any help would be appreciated, or a reference to a similar result. Even an intermediate result would be interesting: for example if $r$ is moreover a fibration, what do we know about $\phi^{!}r$?