Here is the connection between contravariant functors and the opposite category.
Suppose \\(X\\) and \\(Y\\) are categories, and \\(X'\\) is the opposite of category \\(X\\). Suppose \\(T': X' \rightarrow Y\\) is an ordinary functor from \\(X'\\) to \\(Y\\).
Then from \\(T'\\) we can derive a _contravariant_ functor \\(T: X \rightarrow Y\\).
Suppose \\(f: A \rightarrow B\\) in category \\(X\\). Now consider the twin morphism \\(f'\\) in \\(X'\\), with \\(f': B \rightarrow A\\).
Define \\(T(f) = T'(f')\\).
Since \\(T'\\) is a ordinary functor, \\(T'(f'): B \rightarrow A\\).
So \\(T(f): B \rightarrow A\\).
There we see the arrow-reversal, which is required for contravariant \\(T\\).