Query about definition of filtered category

Question

I am entirely new to category theory, and working with the following definition of a filtered category.

A category $I$ is filtered if:

• it is nonempty;

• for each object $x,y\in I$ there is an object $z\in I$ and arrows $x\rightarrow z$ and $y\rightarrow z$;

• for every two arrows $u,v:x\rightarrow y$ there is an arrow $w:y\rightarrow z$ such that $w\circ u=w\circ v$.

My question is basic: is the object $z$ in the third point necessarily the same as the object also called $z$ in the second point?

Answer 1

Purely from a formal stand point, no. The letter $z$ is just a variable, and the two criteria (2) and (3) are not (formally) related.

However, you can take them to be "the same", in the sense that if you have $u, v: x \to y$, then the third criterion guarantees an arrow $w : y \to z$, which means you do get two arrows $x \to z$ and $y \to z$, namely $w \circ u$ and $w$, and then the object $z$ is "the same". Thus, whenever you can "apply" both criteria, you can take the object $z$ to be the same object if you wish. (In this sense the second criterion is implied by the third in the special case where there is an arrow between the objects $x, y$.)

Answer 2

Consider for example the category $$ X \rightrightarrows Y \to Z $$ where everything in this diagram commutes (so there is only one arrow $X \to Z$).

However, as mentioned in another answer, it does not really matter. The point is that we can find such a $Z$ anyway.