Consider $M$ a bicomplete category and $(\operatorname{Fib},\operatorname{Cof},\operatorname{WE})$, $(\operatorname{Fib'},\operatorname{Cof'},\operatorname{WE'})$ be two model structures on $M$ with $\operatorname{Fib}\subseteq \operatorname{Fib'}$, $\operatorname{WE}\subseteq \operatorname{WE'}$. We can deduce $\operatorname{Cof'}\subseteq \operatorname{Cof}$ and from these model structures we can define a new one $(\operatorname{Fib},\operatorname{Cof''},\operatorname{WE'})$ with $\operatorname{Cof''}=\operatorname{LLP}(\operatorname{Fib}\cap \operatorname{WE'})$ if we suppose that $\operatorname{LLP}(\operatorname{Fib})\subseteq \operatorname{LLP}(\operatorname{Fib}\cap \operatorname{WE'})\cap \operatorname{WE'}$ and for any morphism $f$ of $M$ there is a factorization $f=qj$, $q\in \operatorname{Fib}\cap\operatorname{WE'}$, $j\in \operatorname{Cof''}$.
So we have a new model structure and I want :
to know or which pairs of these module structures $id_M$ is a right Quillen functor, and in which it's an equivalence
a concrete example of such a construction
For 1) if we have an adjoint of $id_M$ then it's naturally isomorphic to $id_M$ so I only considered the different modules structures with the adjunction being the trivial one $id_M\dashv id_M$, am I right here ?
For 2) do you have an explicit example ? I know two model categories on topological spaces so I could write things abstractly, that is the left lifting space of the left lifting space intersected with... but it would be nice to have something maybe less interesting but which gives something new and easy to compute.