TITLE: Coequalizers in the Category of Sets
QUESTION [5 upvotes]: This is an elementary question but I cannot get my head around it:
According to the definition of Coequalizers (specifically for the category of Finite Sets), what are the elements of the objects? Different sources talk about equivalence classes and then quotients but they are confusing to understand. Can anyone give a simple example with example Sets and functions even if it is not 100% mathematically right, just to get the idea?

REPLY [5 votes]: In $\Bbb{Set}$, the coequalizer object $Q$ of functions $f,g:X\to Y$ is the quotient set $Y/\sim$ where $\sim$ is the equivalence relation generated by $f(x)\sim g(x)$ for all $x\in X$.
This means that the elements of $Q$ are basically the elements of $Y$, but $f(x)$ and $g(x)$ are regarded the same element in $Q$, for every $x$, i.e. $f(x)=g(x)$ holds in $Q$. [Precisely this would be $q(f(x))=q(g(x))$.]
The equalizer of $f,g$ is the subset of $X$ on which $f$ and $g$ agree ('asking' for those $x\in X$ which make $f(x)=g(x)$).
The coequalizer of $f,g$ is the quotient of $Y$ ('forcing' $f(x)=g(x)$ for all $x\in X$, thus perhaps collapsing some elements together).