TITLE: Unicity up to homotopy of simplicial enrichments
QUESTION [8 upvotes]: 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).

REPLY [6 votes]: If, given any fixed cofibrant object $A$, there is a funtor $map(A,-)$ from $M$ to simplicial sets which preserves weak equivalences between fibrant objects and commutes with homotopy limits up to canonical weak equivalences (e.g. $map(A,-)$ is a right Quillen functor), and such that $\pi_0(map(A,X))=[A,X]$ (functorially) for any fibrant object $X$, then, for any fibrant object $X$, we have a canonical isomorphism $map(A,X)\simeq Map(A,X)$ in the homotopy category of simplicial sets, where $Map(A,X)$ denotes the usual simplicial enrichment. This is because both $map(A,X)$ and $Map(A,X)$ must represent the same presheaf. For (a more precise statement and) a proof, see for instance Remark 6.14 in this paper (but this is in fact a very formal and easy fact which can be found in many ways and disguises in the literature).