Category with coproducts generated by an endomorphism Let's call a category with arbitrary coproducts a $\coprod$-category. A $\coprod$-functor is a functor which preserves coproducts. An example is $\mathsf{Set}$, and this is in fact the universal $\coprod$-category equipped with an object, namely $\{1\}$. By this I mean that for every $\coprod$-category $\mathcal{C}$ there is an equivalence of categories
$$\mathrm{Hom}_{\coprod}(\mathsf{Set},\mathcal{C}) \simeq \mathcal{C},~ F \mapsto F(\{1\}).$$
Question. What is the universal $\coprod$-category equipped with an object $A$ and a morphism $A \to A$? More generally, if $S$ is any set, what is the universal $\coprod$-category equipped with an object and an $S$-indexed family of morphisms $(A \to A)_{s \in S}$?
I am pretty sure that sets are still the objects, resp. the copowers $X \otimes A_{\mathrm{univ}} = \bigoplus_{x \in X} A_{\mathrm{univ}}$. But the morphisms are more complicated.
 A: The category with small coproducts freely generated by a category $\mathcal{C}$ is $\mathbf{Fam} (\mathcal{C})$, the category of families of objects in $\mathcal{C}$. The quickest description is as follows: $\mathbf{Fam} (\mathcal{C})$ is the category obtained by applying the Grothendieck construction to the (contravariant) functor $X \mapsto \mathcal{C}^X$. In other words:


*

*The objects in $\mathbf{Fam} (\mathcal{C})$ are pairs $(X, A)$ where $X$ is a set and $A$ is an object in $\mathcal{C}^X$, i.e., an $X$-indexed family of objects in $\mathcal{C}$.

*The morphisms $(X, A) \to (Y, B)$ in $\mathbf{Fam} (\mathcal{C})$ are pairs $(f, h)$ where $f$ is a map $X \to Y$ and $h$ is a morphism $A \to f^* B$ in $\mathcal{C}^X$, i.e., for each $x \in X$, a morphism $h_x : A_x \to B_{f (x)}$.

*Composition and identities are the obvious ones.


To verify that $\mathbf{Fam} (\mathcal{C})$ has the required universal property, one uses the fact that the canonical comparison map
$$\prod_{x \in X} \coprod_{y \in Y} \mathcal{C} (A_x, B_y) \to \mathbf{Fam} (\mathcal{C}) ((X, A), (Y, B))$$
is a bijection. (This is essentially the same argument that shows that $\mathbf{Ind} (\mathcal{C})$ is the category with filtered colimits freely generated by $\mathcal{C}$.)
Incidentally, when $\mathcal{C}$ is locally small, there is a small-coproduct-preserving functor $\mathbf{Fam} (\mathcal{C}) \to [\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ that extends the Yoneda embedding. The natural bijection above implies this is a fully faithful embedding. The presheaves in the essential image are said to be familially representable. Thus $\mathbf{Fam} (\mathcal{C})$ is equivalent to the full subcategory of $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ spanned by those presheaves that are familially representable, i.e., the presheaves that are small coproducts of representables. (Again, this is entirely analogous to $\mathbf{Ind} (\mathcal{C})$.)
Thus, the category with small coproducts freely generated by an endomorphism is equivalent to a certain full subcategory of the category of sets equipped with an $\mathbb{N}$-action.
