Monoidal relative categories model monoidal ∞-categories

We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal ∞-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal ∞-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.