Let $f(x)\ast y \Leftrightarrow x\ast g(y)$ for some binary relation $\ast$ and functions $f$ and $g$. This is a generalization of Galois connections.
Are things like this studied before?
Note that in my theory $\ast$ is always a symmetric relation (except of the case when this is a Galois connection, and thus $\ast$ is a partial order).
Because it is symmetric, the formula can be rewritten: $f(x)\ast y \Leftrightarrow g(y)\ast x$ (or $y\ast f(x) \Leftrightarrow x\ast g(y)$).
an adjunction in a groupoid.
See http://en.wikipedia.org/wiki/Adjoint_functors for the definition of adjunction and there are notes about how they generalize galois connections