Let us suppose that $I$ is a small category and $\mathcal{E}$ a combinatorial model category. Then there exists two Quillen equivalent combinatorial model category structures on the diagram category $\mathcal{E}^I$, the projective one (fibrations and weak equivalences are pointwise) and the injective one (cofibrations and weak equivalences are pointwise). One can look for instance in Appendix A of *Higher Topos Theory*, Jacob Lurie, for a proof.

I would like to know if it is possible, maybe under some hypotheses on $I$, to obtain the following factorization property that mixes both model structures: any morphism factors through an acyclic projective cofibration followed by an injective fibration.