I'm looking at the Gamma space construction in the paper "Gamma spaces and information" by Matilde Marcolli. For a pointed set $X$ one takes the category $P(X)$ of pointed subsets of $X$ and inclusions, a category $\mathcal{C}$ with coproducts and a zero, and then looks at the functor category $\Sigma_\mathcal{C}(X)$ of summing functors $P(X) \rightarrow \mathcal{C}$. Summing functors send base point of $X$ to zero of $\mathcal{C}$, $\Phi_X(\ast) = 0$, and for $A \cap B = \{\ast\}$, $\Phi_X(A \cup B) = \Phi_X(A) \oplus \Phi_X(B)$, where $\oplus$ denotes the coproduct of $\mathcal{C}$. The Gamma space construction for $X$ is finalized by taking the nerve of the category of summing functors, $\mathcal{N}\Sigma_\mathcal{C}(X)$.

It is then stated that the morphisms in $\Sigma_\mathcal{C}(X)$ are the natural isomorphisms. My question is why this is done? Is there some, possibly basic reason I'm missing, to take only the isomorphisms as morphisms of your category. In this construction or in general. Looking at the answer Category with one isomorphism class, is this somehow related to enforce skeletality on the category and then the nerve above becomes nicer?