If $C$ is a cofibrantly generated model category which is also monoidal biclosed, then to check that $C$ is a monoidal model category, it suffices to check that the Leibniz product of generating cofibrations is a cofibration, that the Leibniz product of a generating cofibration with a generating acyclic cofibration is an acyclic cofibration, and that the unit axiom is satisfied.

The proof is a little involved, so I'd like to have a source to cite this from. Where can I find it in the literature?

I'm happy to assume that the unit is cofibrant.