In category_theory a functor is a mapping between categories.

Given categories \(C\) and \(D\), a functor \(F\) from \(C\) to \(D\) is a mapping that

• associates each object \(X\) in \(C\) to an object \(F(X)\) in \(D\).
• associates each morphism \(f:X \to Y\) in \(C\) to a morphism \(F(f):F(X) \to F(Y)\) in \(D\) such that the following two conditions hold:
  • \(F(id_{X}) = id_{F(X)}\) for every object \(X\) in \(C\).
  • \(F(g \circ f) = F(g) \circ F(f)\) for all morphisms \(f:X \to Y\) and \(g:Y \to Z\) in \(C\).

That is, functors preserve identity morphisms and composition of morphisms