A related idea is that of a contravariant functor, which is defined like a regular functor, except that it maps every arrow to an "arrow going the other way," and it reverses the order of composition.

Here is an example: the powerset functor \$$T\$$, which maps every set \$$S\$$ to its power set \$$2^S\$$ -- all subsets of \$$S\$$ -- and every function \$$f: A \rightarrow B\$$ to the function \$$T(f): 2^B \rightarrow 2^A\$$ which maps a subset of \$$B\$$ to its preimage under \$$f\$$.

So we have \$$T(S) = 2^S\$$, and \$$T(f: A \rightarrow B): T(B) \rightarrow T(A)\$$. There it is, the image of \$$f\$$ goes in the direction opposite to \$$f\$$.

This in turn implies that \$$T\$$ must "reverse the direction of composition." Suppose \$$f: A \rightarrow B\$$, and \$$g: B \rightarrow C\$$. Then the only way composition can work is this:

\$T(f \triangleright g) = T(g) \triangleright T(f)\$