## Epimorphisms are surjective

A morphism $h$ in a category is an \emph{epimorphism} if it is right-cancellative, i.e. for all morphisms $f$, $g$ in the category $f\circ h=g\circ h$ implies $f=g$.

A function $h:A\to B$ is \emph{surjective} (or \emph{onto}) if $B=f[A]=\{f(a): a\in A\}$, i.e., for all $b\in B$ there exists $a\in A$ such that $f(a)=b$.

\emph{Epimorphisms are surjective} in a (concrete) category of structures if the underlying function of every epimorphism is surjective.

If a concrete category has the amalgamation property and all epimorphisms are surjective, then it has the strong amalgamation property1)

1) E. W. Kiss, L. Márki, P. Pröhle, W. Tholen, \emph{Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity}, Studia Sci. Math. Hungar., \textbf{18}, 1982, 79-140 MRreview