Definition 4.20.1. We say that a diagram M : \mathcal{I} \to \mathcal{C} is codirected or cofiltered if the following conditions hold:
the category \mathcal{I} has at least one object,
for every pair of objects x, y of \mathcal{I} there exist an object z and morphisms z \to x, z \to y, and
for every pair of objects x, y of \mathcal{I} and every pair of morphisms a, b : x \to y of \mathcal{I} there exists a morphism c : w \to x of \mathcal{I} such that M(a \circ c) = M(b \circ c) as morphisms in \mathcal{C}.
We say that an index category \mathcal{I} is codirected, or cofiltered if \text{id} : \mathcal{I} \to \mathcal{I} is cofiltered (in other words you erase the M in part (3) above).
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