I am trying to wrap my head around the different ways to describe commutative monoids in categorical algebra.
The Lawvere Theory of commutative monoids can be described as the opposite of the full subcategory $\mathsf{FreeCMon}\subseteq\mathsf{CMon}$ on the commutative monoids of the form $\Bbb N^{\oplus k}$ for $k \geq 0$. This is to say that monoids in a cartesian monoidal category $(\mathcal{C},\times)$ can be described as product preserving functors $\mathsf{CMon}(\mathcal{C}) = \operatorname{Fun}^\times(\mathsf{FreeCMon}^\text{op},\mathcal{C})$.
The free symmetric monoidal category with a commutative monoid is the full subcategory $\mathbf F \subseteq \mathsf{FinSet}$ on the sets $\underline{m} = \{1,...,n\}$. Its symmetric monoidal structure is given by $\underline{m}+\underline{n} = \underline{m+n}$. We obtain commutative monoids in a symmetric monoidal category $\mathcal{C},\otimes)$ as strong monoidal functors via $\mathsf{CMon}(\mathcal{C}) = \operatorname{Fun}^\otimes\big((\mathbf{F},+),(\mathcal{C},\otimes)\big)$.
Consider the full subcategory $\Gamma^\text{op} \subseteq \mathsf{FinSet}_\ast$ on the pointed finite sets $[n] = \{0,...n\}$. For $\mathcal{C}$ pointed and cartesian monoidal, I have often seen special $\Gamma$-objects in $\mathcal{C}$, ie. functors $F:\Gamma^\text{op} \rightarrow \mathcal{C}$, for which the canonical maps $F[k] \rightarrow (F[1])^{\times k}$ are isomorphisms, as models for commutative monoids.
How are these notions related? I arrived at this question by contemplating on how Segal's category $\Gamma$ arises. I found this MO-question here, but I don't think it is answered properly. Especially the connection between free commutative monoids and Segal's $\Gamma$ is elusive to me. Dr. Clough's answer in loc.cit. touches on that, but I don't quite understand, why it suffices to only consider the morphisms $\Bbb N^{\oplus m} \rightarrow \Bbb N^{\oplus n}$, which amount to picking sums of distinct generators.
I think the relation between $\Gamma$ and $\mathbf{F}$ is less surprising. It seems to me that the former is the free pointed symmetric monoidal category with a commutative monoid in the sense that the monoidal semi-cartesian and semi-cocartesian, ie. that the monoidal unit is a zero object. In other words the Segal condition ought to mean that the functor $F$ is strong monoidal with respect to the wedge sum. I am in the process of verifying this, but if someone already knows, I would appreciate a thumbs up or down. Something along these lines is also proposed by Dr. Yuan in a comment on the cited question.
Thank you for your time and help.