Definition. equalizer [leinster2016basic, 5.1.11] [tt-000Q]
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An equalizer of a shape $E(f, g)$ is a fork $(E, \iota )$ over it, such that, for any $(\mathrm {-}, d)$ over the fork, the diagram

$\text{[math]}$ commutes (i.e. any arrow $d : \mathrm {-} \to X$ must uniquely factor through $E$).

For simplicity, we refer to the equalizer of a shape $E(f, g)$ as $\operatorname {Eq}(f, g)$, and $\iota $ is the canonical inclusion.

We say that a category ${\cal C}$ has equalizers iff every shape E in ${\cal C}$ has an equalizer.

Reference. Basic category theory [leinster2016basic]