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