A monoidal Grothendieck construction for $\mathrm{\infty }$-categories

We construct a monoidal version of Lurie’s un/straightening equivalence. In more detail, for any symmetric monoidal $\mathrm{\infty }$-category $\mathbf{C}$, we endow the $\mathrm{\infty }$-category of coCartesian fibrations over $\mathbf{C}$ with a (naturally defined) symmetric monoidal structure, and prove that it is equivalent the Day convolution monoidal structure on the $\mathrm{\infty }$-category of functors from $\mathbf{C}$ to ${\mathbf{C}\mathbf{a}\mathbf{t}}_{\mathrm{\infty }}$. In fact, we do this over any $\mathrm{\infty }$-operad by categorifying this statement and thereby proving a stronger statement about the functors that assign to an $\mathrm{\infty }$-category $\mathbf{C}$ its category of coCartesian fibrations on the one hand, and its category of functors to ${\mathbf{C}\mathbf{a}\mathbf{t}}_{\mathrm{\infty }}$ on the other hand.