Let C be a category and let u : A → B be a morphism in C. Then u is called a monomorphism (or injective) if the map Hom(C, A) → Hom(C,B) which sends v to uv is injective for all objects C in C. Analogously, u is called an epimorphism (or surjective) if the map Hom(B, C) → Hom(A, C) which sends w to wu is injective for all C ∈ C. The morphism u is bijective if u is both injective and surjeetive. An isomorphism, i.e. a morphism having an inverse, is always bijective. The converse is not true in general.
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