We prove that the arrow category of a monoidal model category, equipped withthe pushout product monoidal structure and the projective model structure, is amonoidal model category. This answers a question posed by Mark Hovey, and hasthe important consequence that it allows for the consideration of a monoidalproduct in cubical homotopy theory. As illustrations we include numerousexamples of non-cofibrantly generated monoidal model categories, includingchain complexes, small categories, topological spaces, and pro-categories.
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