Monoidal Relative Categories Model Monoidal $\mathrm{\infty }$-Categories

We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal $\mathrm{\infty }$-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 $\mathrm{\infty }$-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.