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In Awodey's Category Theory on page 62, equalizers are defined like so:

In any category C, given parallel arrows $f, g : A \rightarrow B$, an equalizer of $f$ and $g$ consists of an object $E$ and an arrow $e:E\rightarrow A$, universal such that $f\circ e=g\circ e$.

That is, given any $z:Z\rightarrow A$ with $f \circ z = g \circ z$, there is a unique $u:Z \rightarrow E$ with $e\circ u = z$.

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What does "universal" mean in the first pararaph?

Since the second paragraph starts with "That is", it sounds like the second paragraph is merely clarification, so the "universal" in the first paragraph seems to be doing a lot of work. Or am I misunderstanding?

Edit: I think what I'm looking for is a general definition of "universal" and an explanation of how it applies here, so that the second paragraph becomes obvious.

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    $\begingroup$ Yes, "universal" means that the claim in the second paragraph holds. $\endgroup$ – Kevin Carlson Feb 13 at 19:21
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    $\begingroup$ "That is" is a definition of what it means for it to be universal. $\endgroup$ – Max Feb 13 at 19:24

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