On the invariance of the string topology coproduct

We give a variant of Naef's formula for the failure of invariance of the string topology coproduct under homotopy equivalences, using an obstruction class build from the higher homotopy data one can associate to a homotopy equivalence as well as the ``fake diagonal''. The vanishing of our obstruction class can be seen as a way to measure a form of smallness for homotopy equivalences. We show that the same obstruction rules the failure of invariance for a generalisation of the coproduct to higher dimensional loops.