On the one hand, in their paper Simplicial structures on model categories and functors, Rezk, Schwede and Shipley proved that a simplicial model category structure on a given model category is unique up to simplicial Quillen equivalence.

On the other hand, we know that every model category can be simplicially enriched (for instance taking a cosimplicial resolution of the source or a simplicial one of the target), but in a way which does not give in every case a honest simplicial model category, but simplicial mapping spaces with good homotopy invariance properties and a composition defined up to homotopy (see for instance the category of chain complexes over a field).

Now, let us consider a model category $M$. Suppose that $M$ is equipped with two simplicial mapping space functors $Map(-,-)$ and $\underline{Map}(-,-)$, both homotopy invariant under weak equivalences of a cofibrant source or a fibrant target, and with composition defined at least up to homotopy (so they both induce mapping spaces with a well defined composition on the homotopy category $Ho(M)$ of $M$). Suppose moreover that we have $\pi_0Map(X,Y)\cong[X,Y]\cong\pi_0\underline{Map}(X,Y)$ where $X$ is a cofibrant object of $M$, $Y$ a fibrant object and $[-,-]$ denotes the set of homotopy classes. Do we then have $\pi_nMap(X,Y)\cong\pi_n\underline{Map}(X,Y)$ for every $n$ ?

A more general idea underlying my question is that I wonder if a result similar to the result of Rezk, Schwede and Shipley, in a weaker version, could hold under weaker assumptions (especially, when axiom SM7 of simplicial model categories is not fulfilled anymore).

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