A covariant functor $F:\mathcal C\to \mathcal D$ is a map between categories $\mathcal C$ and $\mathcal D$. It must assign to each object $A$ of $\mathcal C$ an object $F(A)$ an object of $\mathcal D$ and to each morphism $f:A\to B$ of $\mathcal C$ a morphism $F(f):f(A) \to f(B)$ of $\mathcal D$ such that
A contravariant functor “reverses the direction of arrows;” that is, $F$ sends $f:A\to B$ to $F(f): f(B) \to f(A)$ and $F(g\circ f) = F(f) \circ F(g)$.
Wikidata ID: Q864475