∀ {X Y : FintypeCat} (f : X ⟶ Y) (x : (FintypeCat.uSwitch.obj X).obj),
  Y.uSwitchEquiv ((CategoryTheory.ConcreteCategory.hom (FintypeCat.uSwitch.map f)) x) =
    (CategoryTheory.ConcreteCategory.hom f) (X.uSwitchEquiv x)