In the more general (and abstract) setting of category theory, a monomorphism (also called a monic morphism) is a morphismf : X → Y such that
f O g1 = f O g2 implies g1 = g2
for all morphisms g1, g2 : Z → X.
The dual of a monomorphism is an epimorphism (i.e. a monomorphism in a category C is an epimorphism in the dual category Cop).
In the category of sets the monomorphisms are exactly the injective morphisms. Thus the algebraic and categorical notions are the same. The same is true in many other concrete categories such as those of groups, rings, and vector spaces. (Are there any counterexamples?)
There are also useful concepts of regular monomorphism and extremal monomorhpism. A regular monomorphism equalizes some parallel pair of morphisms. An extremal monomorphism is a monomorphism that has no epimorphism as a first factor, unless that epimorphism is an isomorphism.
In the context of universal algebra morphisms are generically known as homomorphisms.
Despite the abstract nature of morphisms, most people's intuition about them (and indeed much of the terminology) comes from the case of concrete categories where the objects are simply sets with some additional structure and morphisms are functions preserving this structure.
Morphisms are often depicted as arrows from their domain to their codomain, e.g.
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