Epimorphisms are surjective

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A morphism h in a category is an epimorphism if it is right-cancellative, i.e. for all morphisms f, g in the category foh = goh implies f = g.

A function h : A → B is surjective (or onto) if B = f[A] = {f(a) | a ∈ A}, i.e., for all b ∈ B there exists a ∈ 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 property.

[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]