| context | ${\bf C}$ … category |
| definiendum | $f \in\mathrm{it} $ |
| inclusion | $f:{\bf C}[A,B]$ |
| postulate | $\langle B,\prod_{B}1_A\rangle$ … pushout of $f$ along itself |
See Monomorphism.
In ${\bf{Set}}$ the epimorphisms are the surjections. But people like to point out that in general, epis are quite different from surjections and also more difficult to classify (as opposed to monos, which mostly behave exactly like injections). See the nLab link below for variations (or rather further restrictions) of the concept.
todo