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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.

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# Evaluation functors and injective model structure on diagram categories

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 !