On the tensor product of enriched $\infty$-categories

Abstract

We show that the tensor product of $\infty$-categories enriched in a suitable monoidal $\infty$-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 $\infty$-categories, and that the tensor product and internal hom are compatible with this.