The Stacks project

Lemma 4.19.5. Let $\mathcal{I}$ be an index category, i.e., a category. Assume that for every pair of objects $x, y$ of $\mathcal{I}$ there exist an object $z$ and morphisms $x \to z$ and $y \to z$. Let $M : \mathcal{I} \to \textit{Ab}$ be a diagram of abelian groups over $\mathcal{I}$. Then the colimit of $M$ in the category of sets surjects onto the colimit of $M$ in the category of abelian groups.

Proof. Recall that the colimit in the category of sets is the quotient of the disjoint union $\coprod M_ i$ by relation, see Section 4.15. Similarly, the colimit in the category of abelian groups is a quotient of the direct sum $\bigoplus M_ i$. The assumption of the lemma means that given $i, j \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and $m \in M_ i$ and $n \in M_ j$, then we can find an object $k$ and morphisms $a : i \to k$ and $b : j \to k$. Thus $m + n$ is represented in the colimit by the element $M(a)(m) + M(b)(n)$ of $M_ k$. Thus the $\coprod M_ i$ surjects onto the colimit. $\square$

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