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I have added to coequalizer basic statements about its relation to pushouts.
In the course of this I brought the whole entry into better shape.
Something is wrong with the terminology in the idea section.
the projection function $p \colon Y \longrightarrow Y/_\sim$ satisfies
$p \circ f = p \circ g$and in fact $p$ is universal with this property, hence it “co-equalizes” $f$ and $g$.
In the standard terminology, one says that $p$ coequalizes a parallel pair $f,g$ if $p\circ f = p\circ g$, period. No universality. (Co)equalizing is the same as making a (co)cone here, not the same as being a (co)equalizer/universal (co)cone !
I agree. It should read to say, “$p$ is the coequalizer of the maps $f, g$”. Edit: I made an adjustment there, and also changed the word “projection” to “quotient” since projection is given the specific meaning having to do with products.
Added this:
Coequalizers were defined in the paper
for any finite collection of parallel morphisms. The paper refers to them as right equalizers, whereas equalizers are referred to as left equalizers.
added pointer to:
added missing link back to regular epimorphism
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