Why does morphism in equalizer have to be monomorphism? I'm watching this lecture And the lector says that p have to be monomorphism. I've read wikipedia#In_category_theory) and it also mentioned there, but no proof is given. If we use notation from wikipedia article I get that `u` is unique and because of that we have a situation that is similar to monomorphism, but in monomorphism any two morphisms from `O` to `E` that compose with `eq` would have to be equal, not only those which factorize some other morphism.

Note that $eq$ itself equalize $f$ and $g$; thus if $eq\circ v=eq\circ w$, you can define $m=eq\circ v$, and then $$f\circ m=f\circ eq\circ v=g\circ eq \circ v=g\circ m.$$ The universal property implies that there must be a unique $u$ such that $eq\circ u=m$; but both $v$ and $w$ have this property, thus $u=v=w$.