  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 $\infty$-categories to $\mathcal W$-enriched $\infty$-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 $\infty$-category to a category expressible in
diagrammatic terms, we give a criterion for it to take $(\mathcal{O} \otimes
\mathcal{E})$-monoidal $\infty$-categories to $\mathcal{O}$-monoidal
$\infty$-categories using a "pointwise" monoidal structure.
