Consider the well-known categorical result:
Theorem 1.5.9 (of Riehl's book). A functor defining an equivalence of categories is full, faithful, and essentially surjective on objects. Assuming the axiom of choice, any functor with these properties defines an equivalence of categories.
I was wondering if the formal counterpart of this result in multicategories is also true. For a definition of a multicategory, a multifunctor and a natural transformation of multifunctors, see this, pp. 1, 2, 4, respectively. Define an isomorphism of multifunctors to be a natural transformation of multifunctors whose components are all isomorphisms. Now one can generalize Definition 1.5.4 of Riehl's book to
Definition. An equivalence of multicategories consists of multifunctors $F:\mathsf{C}\leftrightarrows\mathsf{D}:G$ together with natural isomorphisms $\eta:1_\mathsf{C}\cong GF$, $\epsilon:FG\cong 1_\mathsf{D}$.
So under all of these definitions, my first question is: does Theorem 1.5.9 generalize to multicategories? Or is there something that may fail? I don't have the time right now to delve into higher abstractions by trying to generalize myself Riehl's proof. So my second question is: is the proof already written in some book or does the result show up somewhere in the literature?