Lax monoidality for products of enriched higher categories

We prove that a lax ${\mathbb{E}}_{n+1}$-monoidal functor from $\mathcal{V}$ to $\mathcal{W}$ induces a lax ${\mathbb{E}}_n$-monoidal functor from $\mathcal{V}$-enriched $\mathrm{\infty }$-categories to $\mathcal{W}$-enriched $\mathrm{\infty }$-categories in the sense of Gepner–Haugseng. We prove this as part of a general-purpose interaction with the Boardman–Vogt tensor product $\otimes$: given a construction that takes an $\mathcal{E}$-monoidal $\mathrm{\infty }$-category to a category expressible in diagrammatic terms, we give a criterion for it to take $(\mathcal{O}\otimes \mathcal{E})$-monoidal $\mathrm{\infty }$-categories to $\mathcal{O}$-monoidal $\mathrm{\infty }$-categories using a “pointwise” monoidal structure.