TITLE: Special $\Gamma$-categories and symmetric monoidal categories
QUESTION [8 upvotes]: 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 pseudomonoid 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).

REPLY [3 votes]: Tom Leinster's book is very old. For higher category theory 2003 is like a previous epoch. In those times there were many competing definitions of higher category theory and higher algebra, for most of them it was unclear if they are equivalent or whether they even model all homotopy types. For some of them this was eventually proven false, e.g. the once-favoured derivators are unsufficient to properly work with higher algebra and the strict $\omega$-categories are obviously unsufficient for the homotopy hypothesis (homotopy types = $\infty$-groupoids). The main delicate points are in giving the proper definitions and handling all coherence conditions, so that the theory is flexible enough for all purposes but at the same time is still tractable. For this reason the aforementioned statement is stated not as a theorem (probably it is so in Leinster's foundations, but I don't have the print copy at hand) but rather as a guiding principle which should be true in any proper higher category theory. This is certainly true in the now-accepted foundations via quasicategories and higher Segal spaces. In fact, it is usually taken as the definition of a monoidal higher category, so the only question is whether it covers all classical example (it does). J. Lurie's "Higher Algebra" is nowadays the go-to reference for all questions about the foundations of higher algebra, and his "Higher Topos Theory" lays the used foundations of $(\infty, 1)$-category theory.
If you are working within classical 1-category theory, then the statement is very easy to prove explicitly. I am unaware of any specific reference for it.  For $A_\infty$-monoids in homotopy theory I believe that the approach via $\Gamma$-spaces originates from the work of G. Segal. You can check the references in the relevant page on nLab.
Looking at the arXiv version of Leinster's paper, it also doesn't really discuss $\infty$-categories, nor does it even define an equivalence of $\omega$-categories (see par. 9.2), so the equivalence between monoidal categories and $\Gamma$-spaces was likely never really proved in that foundations.