Let us assume that I have two sets of combinatorial objects, $A$ and $B$, and I am looking for a bijection (in particular a map) $\psi:A \to B$ between these sets, usually required to preserve some additional statistic.
For example, a combinatorial proof of the qt-symmetry of the qt-Catalan numbers can be stated in this form, and this is still open. Same can be said about the GOG-MAGOG problem.
Suppose now that $B$ has a lot of structure, perhaps with an action of $S_n$ on it, or similar. To find $\psi$, one approach would then be to look for a similar action on $A$, and then require that the maps commute. This of course adds a lot more restrictions on $\psi$, making the search space smaller.
Usually, articles usually only presents $\psi$, not how it was found. So my question is, has this line of thinking been used explicitly somewhere?
Have you used this "functor"-method to find bijections?
I suppose that inventing a new statistic on $B$, say $\sigma: B \to \mathbb{N}$, and then looking for the corresponding statistic on $\tau:A \to \mathbb{N}$, such that $ \sigma \circ \phi = \tau$, also counts as a "functorial" approach, but this is more commonly known as refinements or q-analogues, and not what I am looking for.
