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. We provide details of the proof of Lurie's theorem on operadic Kanextensions. Along the way, we generalize the construction of monoidal envelopesof $\infty$-operads to families of $\infty$-operads and use it to construct thefiberwise direct sum functor, both of which we characterize by certainuniversal properties. Aside from their uses in the proof of Lurie's theorem,these results and constructions have their independent interest.

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