Epimorphisms are surjective

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

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, Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity, Studia Sci. Math. Hungar., 18, 1982, 79-140 MRreview