Monoidal Relative Categories Model Monoidal $\infty$-Categories

Abstract: We show that the homotopy theory of monoidal relative categories is equivalent to that of monoidal $\infty$-categories, as well as its symmetric monoidal version. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal $\infty$-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus--Sagave.