Opposite categories and contravariance

Motivation: Consider the situation, in an arbitrary category, $$\leftcat{X' \buildrel u \over \to X} \buildrel f \over \to \rightcat{Y \buildrel v \over \to Y'}$$ Often we want to consider $\leftcat u$ and $\rightcat v$ as operators operating on $f$.
But then consider the slightly more complex situation $$\leftcat{X'' \buildrel u' \over \to X' \buildrel u \over \to X} \buildrel f \over \to \rightcat{Y \buildrel v \over \to Y' \buildrel v' \over \to Y''}$$ Comparing letting first $\leftcat u$ and $\rightcat v$ operate on f, then their primed relatives,
to first composing the operators, then letting the composed operators operate on f,
we have $$\leftcat{u'}(\leftcat u f \rightcat v)\rightcat{v'} = \leftcat{(u'u)} f \rightcat{(vv')}$$