To complete the demonstration that \\(T\\) is a contravariant functor, it only remains to show that it reverses the order of composition:

\\[T(f \triangleright g) = T'((f \triangleright g)') = T'(g' \triangleleft f') = T'(g') \triangleright T'(f') = T(g) \triangleright T(f)\\]

Note: we only used \\(\triangleleft\\) in one place here, only as a stylistic reminder that composition is talking place in an opposite category.