The following definitions are borrowed from Grothendieck's SGA 4, p.14-15.
Definition 1 A category $I$ is called pseudo-filtered if it satisfies the following conditions.
1) For every two morphisms with common domain $f\colon i \rightarrow j$ and $g\colon i \rightarrow j'$, there exists an object $k$ and two morphisms $u\colon j \rightarrow k$ and $v\colon j' \rightarrow k$ such that $u\circ f = v\circ g$.
2) For every two parallel morphisms $u,v\colon i\rightarrow j$ in $I$, there exists an object $k$ and an morphism $w\colon j\rightarrow k$ such that $w\circ u = w\circ v$.
Definition 2 A category $I$ is called filtered if it is pseudo-filtered, nonempty and connected.
My question Is the above definition of filtered categories equivalent to the usual one? If yes, how do you prove it?
It is clear that a filtered category is a nonempty connected pseudo-filtered category. We will prove the converse. Let $I$ be a nonempty connected pseudo-filtered category. Let $i, j$ be objects of $I$. Since $I$ is connected, there exist $n$ arrows connecting $i$ and $j$. It suffices to prove that there exist arrows $i \rightarrow k$ and $j \rightarrow k$. It is easy to prove this using induction on $n$ and the condition 1).