Let $\mathscr{C}$ be a category of fibrant objects. For objects $X$, $Y$ of $\mathscr{C}$ we consider the category $\underline{\text{Hom}}(X,Y)$ of spans $X\leftarrow X'\rightarrow Y$, where $X'\to X$ is a trivial fibration. We can consider the full subcategory $\underline{\text{Hom}}'(X,Y)$, consisting of spans where the morphism $X'\to Y$ is a fibration.

I seem to have convinced myself, that $\underline{\text{Hom}}'(X,Y)\to\underline{\text{Hom}}(X,Y)$ induces a homotopy equivalence on geometric realizations. In particular, I can restrict to the subcategories $\underline{\text{Hom}}'(X,Y)$ to get the correct simplicial localization of $\mathscr{C}$.

Maybe someone knows that this is impossible? (The proof would go via constructing functorial fibrant resolutions for morphisms with target $Y$ using a fixed path object for $Y$.)