Monoidal Envelopes of Families of $\mathrm{\infty }$-Operads and $\mathrm{\infty }$-Operadic Kan Extensions

We provide details of the proof of Lurie’s theorem on operadic Kan extensions. Along the way, we generalize the construction of monoidal envelopes of $\mathrm{\infty }$-operads to families of $\mathrm{\infty }$-operads and use it to construct the fiberwise direct sum functor, both of which we characterize by certain universal properties. Aside from their uses in the proof of Lurie’s theorem, these results and constructions have their independent interest.