On the tensor product of enriched $\mathrm{\infty }$-categories

We show that the tensor product of $\mathrm{\infty }$-categories enriched in a suitable monoidal $\mathrm{\infty }$-category preserves colimits in each variable, fixing a mistake in an earlier paper of Gepner and the author. We also prove that essentially surjective and fully faithful functors form a factorization system on enriched $\mathrm{\infty }$-categories, and that the tensor product and internal hom are compatible with this.