A covariant functor F, between two categories C and D, is an assignment to each object X of C an object F(X) of D and to each morphism f:X→Y of C a morphism F(f):F(X)→F(Y) satisfying F(1_X) = 1_F(X) and F(g∘f) = F(g)∘F(f). The functor is contravariant if, instead, F(g∘f) = F(f)∘F(g). Assigning every real vector space to its dual space, and every linear map to its dual map, is a contravariant functor on the category of real vector spaces.