Generally speaking, a homomorphism is a structure-preserving mapping 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:
A monomorphism is an injective homomorphism.
An epimorphism is a surjective homomorphism.
An isomorphism is a bijective homomorphism.
An endomorphism is a homomorphism from a set to itself.
An automorphism is a bijective endomorphism.
The sets of all homomorphisms, monomorphisms, epimorphisms, isomorphisms between $X$ 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, automorphisms on $X$ are respectively denoted by
\[\endo X, \quad \aut X.\]$A$ and $B$ are isomorphic, written
\[A \simeq B,\]if an isomorphism between them exists.