Preservation of coproducts by coproducts

Question

Let $\mathcal{C}$ be a category with (small) coproducts, $F, G : S \to \mathcal{C}$ functors (here $S$ seen as a discrete category). Is the following true? $$\coprod\limits_{c \in S} (F(c) + G(c)) \cong \coprod\limits_{c \in S} F(c) + \coprod\limits_{c \in S} G(c) $$ What is the standard way of approaching this kind of isomorphisms?

Answer 1

Yes, this is true, and the standard way to prove it is to use the Yoneda Lemma. For any object $X$ of $\mathcal{C}$, the universal property of coproducts gives you a chain of isomorphisms \begin{align}\operatorname{Hom}_\mathcal{C}\left(\coprod\limits_{c \in S} (F(c) + G(c)),X\right) & \cong \prod_{c\in S}\operatorname{Hom}_\mathcal{C}(F(c)+G(c),X)\\ & \cong \prod_{c\in S}\operatorname{Hom}_\mathcal{C}(F(c),X)\times \operatorname{Hom}_\mathcal{C}(G(c),X)\\ & \cong \left( \prod_{c\in S}\operatorname{Hom}_\mathcal{C}(F(c),X)\right)\times\left(\prod_{c\in S} \operatorname{Hom}_\mathcal{C}(G(c),X)\right)\\ & \cong \operatorname{Hom}_\mathcal{C}\left(\coprod\limits_{c \in S} F(c),X\right)\times \operatorname{Hom}_\mathcal{C}\left(\coprod\limits_{c \in S} G(c),X\right) \\ & \cong\operatorname{Hom}_\mathcal{C}\left(\coprod\limits_{c \in S} F(c) + \coprod\limits_{c \in S} G(c),X\right)\end{align} which are all natural with respect to $X$, so by the Yoneda Lemma the isomorphism between the first and last term must be induced by an isomorphism $$\coprod\limits_{c \in S} (F(c) + G(c)) \cong \coprod\limits_{c \in S} F(c) + \coprod\limits_{c \in S} G(c).$$

In fact, you can prove in pretty much the same way that in any category, "(co)limits commute with (co)limits" (when they exist).