An equalizer in a category $\mathcal{C}$ is a couple $(E,eq)$ consisting in an object $E$ and a morphism $eq:E\longrightarrow X$ so that $f\circ eq=g\circ eq$ for every pair of parallel morphisms $f,g:X\longrightarrow Y$ and if for every other object $O$ and morphism $m:O→X$ there exists a unique morphism $u:O→E$ so that $eq\circ u=m$.

In the category $Set$, by taking sets and functions between them, an equalizer is the set of elements of the common domain where the functions are equal, that is: $Eq(f,g)=\{x\in X/f(x)=g(x)\}$ with $X$ a set and $(f,g)$ a couple of parallel morphism in Set.

My question is: can we say that the equalizer set is minimal among all the *equalizer sets* (like $O$ in the definition)?

Thanks for participate.

`$m$`

should also satisfy`$f\circ m=g\circ m$`

. Sorry. Anything new then? – Doctor Gibarian Jan 31 '10 at 12:01ofa parallel pair; in particular, the "for every" in the first sentence of the original post is incorrect – Yemon Choi Jan 31 '10 at 19:49