Let $A$ be a small simplicial category. The category $Fun(A,s\mathrm{Set})$ of simplicial functors from $A$ to simplicial sets can be given the projective model structure in which fibration and weak equivalences are objectwise.


Now assume further that $A$ is a symmetric monoidal category with respect to some binary operarion $\circ: A\times A\to A$. We can define the Day (or convolution) tensor product on $Fun(A,s\mathrm{Set})$ by the following coend:
$$F\otimes G(a)=A(-\circ -,a)\otimes\_{A\times A}F(-)\times G(-)$$


My question is:


Is it true that the projective model structure is a monoidal model category in the sense that it satisfies the pushout-product axiom and if so is there a place where this is written down ?



Monoidal model category structure on a functor category.
[This should probably be a comment, since it is so short. Nevertheless, it is an answer to the question.]


The result you ask for is a consequence of proposition 2.2.15 in [Sam Isaacson's Ph.D. thesis](http://www.math.uwo.ca/~sisaacso/PDFs/diss.pdf) (Harvard University, 2009). That proposition is stated for combinatorial symmetric monoidal (closed) model categories, and not just for simplicial sets. In this general case, the statement in Sam Isaacson's thesis requires the base category to have *virtually cofibrant* morphism objects. This condition is automatic for simplicial sets, as all simplicial sets are cofibrant in the usual model structure.


I have no idea who first proved or published the above result.


