In Model Categories and Their Localizations, Definition 11.7.2, for $M$ a simplicial cofibrantly generated model category, $C$ a small category, Hirschhorn gives the following simplicial structure on $M^C$: \begin{align*} F \otimes K(a) &= F(a) \otimes K \\ \mathrm{Map}(F,G)_n &= M^C(F \otimes \Delta[n],G) \end{align*} And says in Theorem 11.7.3 that having composition pointwise for the Map-spaces gives $M^C$ the structure of a model category with the projective model structure. But I don't see why taking composition pointwise will give a map of diagrams. It seems that it depends on some sort of naturality of the composition of Map-spaces, something that we can make precise by the following proposition:
Designating the composition at level $n$ of the Map-spaces by $\circ_n$, with $f : A \to A'$, $g : B \to B'$, $h : C \to C'$ three morphisms in $M$, and the following diagram:
If the two small rectangles commute, then the large rectangle should commute as well. It seems that Hirschhorn does not prove this. If it is true, how can we prove it? And if it is false, what did I not understand about Hirschhorn's construction?