  Abstract. 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 $\infty$-operads to families of $\infty$-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.
