I'm studying a small category $A$ and diagrams of based spaces or spectra indexed by $A$ (so let's say diagrams in a category $C$ that's closed symmetric monoidal, has a compatible model structure, etc.). I'm told that in this setting there's a projective model structure on diagrams $A \rightarrow C$, where the fibrations and weak equivalences are defined objectwise; and an injective model structure, where the cofibrations and weak equivalences are defined objectwise.


I want to take two diagrams $X,Y$ and form a mapping space $\text{Map}(X,Y)$ in a homotopy-invariant way. (The mapping space should be a subspace of the product of the mapping spaces for each level. In a more formal setting, I can pass to the twisted arrow category on $A$ and take an equalizer.) I should be able to accomplish this by taking a cofibrant replacement of $X$ and a fibrant replacement of $Y$, but there are two model structures available for doing this. This gives me two different mapping spaces; I want to show that they are equivalent. Any suggestions for approaches/tools/references?


(An intermediate step might be to show that they are both equivalent to the space I get by taking a projective-cofibrant replacement of $X$ and an injective-fibrant replacement of $Y$.)



mapping spaces of diagrams
Let me mention yet another approach using the simplicial localization due to Dwyer and Kan.


Given any category $\mathcal{C}$ equipped with a class of morphisms $W$ one can form a simplicial category $L(\mathcal{C}, W)$ which depends only on $\mathcal{C}$ and $W$ and not on any auxiliary structure like model structure. However, it is true that if $\mathcal{C}$ comes with a model structure (with weak equivalences $W$), then the mapping spaces of $L(\mathcal{C}, W)$ are weakly equivalent to the mapping spaces constructed using (co)simplicial resolutions coming from the model structure. In particular, if we have two model structures on $\mathcal{C}$ with the same weak equivalences $W$ (as in your situation), then the mapping spaces constructed from those model structures are weakly equivalent.


The precise reference is Proposition 4.4 of *Function Complexes in Homotopical Algebra* by Dwyer and Kan.


