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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.
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