Functor

A mapping between categories that assigns to each object and morphism in the source category an object and morphism in the target category while preserving identities and composition (covariant) or reversing arrows (contravariant via the opposite category).

A mapping F between categories C and D that assigns to each object X of C an object F(X) of D and to each morphism f: X→Y in C a morphism F(f): F(X)→F(Y) in D, preserving identities (F(id_X)=id_{F(X)}) and composition (F(g◦f)=F(g)◦F(f)).

A mapping between categories that assigns to every object of a source category an object of a target category and to every morphism in the source a morphism in the target, preserving identity morphisms and composition.