Let $\mathcal{C}$ be a category with (small) coproducts, $S$, $S'$ sets, and $F : S + S' \to \mathcal{C}$ a functor (here $S + S'$ seen as a discrete category). Is the following true? $$\coprod\limits_{c \in S + S'} F(c) \cong \coprod\limits_{c \in S} F(\iota_0(c)) + \coprod\limits_{c \in S'} F(\iota_1(c)) $$ where $\iota_i$ are the injections.
I am a bit confused, since I remember this kind of wild reasoning was forbidden for series of numbers?