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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?

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  • $\begingroup$ Yes, this is true. This is not a series of numbers. $\endgroup$ – Qiaochu Yuan Jan 18 '18 at 10:04
  • $\begingroup$ Perhaps this is too much for asking on a comment... but is there any intuition on why this kind of "possibly infinite" colimits works so differently to a series? Also, any hint on the proof? $\endgroup$ – M. Learner Jan 18 '18 at 10:06
  • $\begingroup$ It is simply not like a series. You don't have to take anything like a limit (in the analysis sense), and unlike series, coproducts can always be taken in any order. The proof is very simple: ask yourself what a map out of each side is and verify they have the same universal property (then use the Yoneda lemma). There is nothing wild about the reasoning and the infinite and finite cases are identical. $\endgroup$ – Qiaochu Yuan Jan 18 '18 at 10:07
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    $\begingroup$ But if you insist... the kinds of things that go wrong for series involve negative numbers, but if you imagine $C$ as being like the category of sets then all the "numbers" involved are nonnegative. For series of nonnegative reals this is true, if $\infty$ is one of the allowable values. $\endgroup$ – Qiaochu Yuan Jan 18 '18 at 10:08

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