Definition 1. Broadly speaking, a homomorphism is a structure-preserving function between two structured sets. The precise definition of a homomorphism depends on the context.
Once a homomorphism is defined, the following concepts are automatically understood:
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A monomorphism is an injective homomorphism.
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An epimorphism is a surjective homomorphism.
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An isomorphism is a bijective homomorphism.
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An endomorphism is a homomorphism from a set to itself.
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An automorphism is a bijective endomorphism.
The sets of all homomorphisms, monomorphisms, epimorphisms and isomorphisms between two sets $X$ and $Y$ are respectively denoted by
\[\hom(X, Y), \quad \mon(X, Y), \quad \epi(X, Y), \quad \iso(X, Y).\]The sets of all endomorphisms and automorphisms on a set $X$ are respectively denoted by
\[\endo X, \quad \aut X.\]