Let $\Gamma^{op}$ be the category of finite pointed sets. A special $\Gamma$-category is a functor $Y:\Gamma^{op}\to Cat$ such that the canonical maps $Y[n]\to Y[1]^n$, called *Segal maps*, are equivalences of categories, for all $n\geq 0$.

There is a canonical way of associating to an unbiased symmetric monoidal category a special $\Gamma$-category (called a “homotopy monoidal category” by Tom Leinster in Higher Operads, Higher Categories).

I think that Moritz Groth, in his Example 3.4 of his course in infinity categories, is saying that this association is an equivalence.

However, Tom Leinster, pages 120-121 of the aforementioned book, says that “there is reasonable hope” that this is true.

I would like to know whether this result is true or not, and where do the delicate points lie. If it is true, I what are some references where the result is proven in detail?

EDIT: I would like to add some context. A commutative monoid in a cartesian monoidal category $\mathcal C$ is, equivalently, a functor $\Gamma^{op}\to \mathcal C$ that satisfies that the Segal maps are isomorphisms.

My point is that I would like to apply this to the concept of symmetric monoidal category itself. However, a (small) symmetric monoidal category is not monoid, but a symmetric *pseudo*monoid in the cartesian 2-category $Cat$.

So I guess I would like to know whether it is true that symmetric pseudomonoids in 2-categories $D$ are equivalently pseudofunctors $\Gamma^{op} \to D$ that satisfy that the canonical maps $Y[n]\to Y[1]^n$ are 1-equivalences. Note that this is not exactly what I wrote in the first line (here I'm letting $Y$ be a pseudofunctor).