If $M$ is a combinatorial model category, it's known by the experts that there is a ''natural'' model structure on diagram categories $Hom(C,M)$, which is the **projective model structure**. The fibrations and weak equivalences are defined point wise. 

There is also the one called **injective model structure**, where the cofibrations and weak equivalences are defined point wise. 

I would like to know if for a given a morphism $\alpha \in Arr(C)$, the evaluation at $\alpha$ can be a right Quillen functor with the **injective model structures on each side**:

$Ev_\alpha: Hom(C,M) \to M^2$ 

Thanks ! 


Edit: Here $M^2= Hom([0 \to 1], M)= Arr(M)$, sorry for the confusion.