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.



Evaluation functors and injective model structure on diagram categories
As already observed, this is not always true, but I will give a more general sufficient condition, which I believe may also be necessary. The condition is that $\alpha$ is an epimorphism in $C$.


I will denote the one arrow category by $[1]$. The left adjoint to $\mathrm{ev}\_\alpha : \mathcal{M}^C \to \mathcal{M}^{[1]}$ is the left Kan extension $\mathrm{Lan}\_\alpha : \mathcal{M}^{[1]} \to \mathcal{M}^C$. It can be computed explicitly. If $X \in \mathcal{M}^{[1]}$ and $c \in C$, then $(\mathrm{Lan}\_\alpha X)\_c$ is the pushout of $C(\alpha\_1, c) \times X\_0 \to C(\alpha\_1, c) \times X\_1$ along $C(\alpha\_1, c) \times X\_0 \to C(\alpha\_0, c) \times X\_0$. If $\alpha$ is an epimorphism, then $C(\alpha\_1, c) \to C(\alpha\_0, c)$ is injective for all $c$ and the pushout in question is a pushout along a cofibration. It follows from the Gluing Lemma that $\mathrm{Lan}\_\alpha$ preserves levelwise (acyclic) cofibration, so it is a left Quillen functor.


