The Stacks project

Lemma 4.19.8. Let $\mathcal{I}$ be an index category, i.e., a category. Assume

  1. for every pair of morphisms $a : w \to x$ and $b : w \to y$ in $\mathcal{I}$ there exist an object $z$ and morphisms $c : x \to z$ and $d : y \to z$ such that $c \circ a = d \circ b$, and

  2. for every pair of morphisms $a, b : x \to y$ there exists a morphism $c : y \to z$ such that $c \circ a = c \circ b$.

Then $\mathcal{I}$ is a (possibly empty) union of disjoint filtered index categories $\mathcal{I}_ j$.

Proof. If $\mathcal{I}$ is the empty category, then the lemma is true. Otherwise, we define a relation on objects of $\mathcal{I}$ by saying that $x \sim y$ if there exist a $z$ and morphisms $x \to z$ and $y \to z$. This is an equivalence relation by the first assumption of the lemma. Hence $\mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ is a disjoint union of equivalence classes. Let $\mathcal{I}_ j$ be the full subcategories corresponding to these equivalence classes. The rest is clear from the definitions. $\square$

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