Covariant functor, identity functor, composite functor, and contravariant functor

For categories $\mathcal C,\mathcal D$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), a covariant functor $F:\mathcal C\to\mathcal D$ assigns an object $FA$ to every object $A$ and a morphism $Ff:FA\to FB$ to every $f:A\to B$, satisfying

$F(1_A)=1_{FA},\qquad F(g\circ f)=Fg\circ Ff.$

The identity functor acts identically on objects and morphisms. For $F:\mathcal C\to\mathcal D$ and $G:\mathcal D\to\mathcal E$, the composite functor $GF$ has $(GF)A=G(FA)$ and $(GF)f=G(Ff)$.

A contravariant functor from $\mathcal C$ to $\mathcal D$ means a covariant functor $\mathcal C^{\mathrm{op}}\to\mathcal D$, using Opposite category $\mathcal C^{\mathrm{op}}$.