${\mathbb{E}}_n$-algebras in m-categories

We prove a connectivity bound for maps of $\mathrm{\infty }$-operads of the form ${\mathbb{A}}_{k_1}\otimes \cdots \otimes {\mathbb{A}}_{k_n}\to {\mathbb{E}}_n$, and as a consequence, give an inductive way to construct ${\mathbb{E}}_n$-algebras in $m$-categories. The result follows from a version of Eckmann-Hilton argument that takes into account both connectivity and arity of $\mathrm{\infty }$-operads. Along the way, we prove a technical Blakers-Massey type statement for algebras of coherent $\mathrm{\infty }$-operads.