Definition 7.2.4.1. Let $\operatorname{\mathcal{C}}$ be a category. We say that $\operatorname{\mathcal{C}}$ is filtered if it satisfies the following conditions:
The category $\operatorname{\mathcal{C}}$ is nonempty.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there exists an object $Z \in \operatorname{\mathcal{C}}$ and a pair of morphisms $u: X \rightarrow Z$ and $v: Y \rightarrow Z$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ and every pair of morphisms $f_0, f_1: X \rightarrow Y$, there exists a morphism $v: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$ satisfying $v \circ f_0 = v \circ f_1$.