Definition:Filtered Category

Definition
Let $\mathbf I$ be a category.

$\mathbf I$ is filtered, when:
 * $\mathbf I$ is not empty.
 * For every two objects $i$ and $i'$ of $\mathbf I$, there exists an object $k$ and morphisms $f: i \to k$ and $f' : i' \to k$.
 * For every two objects $i$ and $j$ of $\mathbf I$ and every two morphisms $u : i \to j$ and $v : i \to j$, there exists an object $k$ a morphism $w : j \to k$, such that $w \circ u = w \circ v$.