How to prove $\text{Map}_C(X,Y)$ is a grouplike commutative monoid object of the $\infty$-category of spaces?

Question

For an additive $\infty$-category $C$ every object $Y$ is a commutative grouplike object in $C$. Now my question is how we can show it is the case for any mapping space $\text{Map}_C(-,Y)$ in the $\infty$-category of spaces? Is there any reference for that?

Maybe I should mention that there is an analog case in ordinary categories. I mean for any additive ordinary category $A$ and any object $X$ in it, we know the functor $\text{Hom}_A(-,X)$ can uniquely be lifted to a functor from $A$ to the category of $\text{Mod}(\mathbb{Z})$ of $\mathbb{Z}$-modules (Abelian groups).

Answer 1

A reference is Corollary II.17 of the lecture notes on algebraic and hermitian K theory by Fabian Hebestreit (typeset copy by Ferdinand Wagner available here). The argument is relatively short, so I have summarized it below.

Roughly, for each $X : \mathcal{C}$, it is clear that $\hom(X,-)$ is a finite product preserving functor and so induces a functor $\hom(X,-)_* : \mathrm{CGrp}(\mathcal{C}) \to \mathrm{CGrp}(\mathcal{S})$. We can precompose this by an equivalence $\mathrm{CGrp}(\mathcal{C}) \simeq C$ to obtain a functor $\hom(X,-) : C \to \mathrm{CGrp}(\mathcal{S})$. The problem is to extend this to a functor in two arguments $\mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathrm{CGrp}(\mathcal{S})$.

For this, it suffices to note that the equivalence $\mathrm{CGrp}(\mathcal{C}) \simeq \mathcal{C}$ induces a map:

$$\mathcal{C}^{\mathrm{op}} \to (\mathcal{C} \to (\mathrm{Fin}_* \to \mathcal{S}))$$

This sends $X : \mathcal{C}^{\mathrm{op}}$ to the functor $\hom(X,-)_* : (\mathrm{Fin}_* \to \mathcal{C}) \to (\mathrm{Fin}_* \to \mathcal{S})$ after restricting the domain to $\mathcal{C} \simeq \mathrm{CGrp}(\mathcal{C}) \subseteq \mathrm{Fin}_* \to \mathcal{C}$. This is clearly functorial, and so it suffices to argue that after restricting the domain, $\hom(X,-)_*$ lands in the full subcategory $\mathrm{CGrp}(\mathcal{S})$. However, this is precisely what we have shown above: $\hom(X,-)_*$ sends $\mathbb{E}_\infty$-groups to $\mathbb{E}_\infty$-groups.