  The canonical (CQG) and asymptotically safe (ASQG) approach to quantum
gravity share to be both non-perturbative programmes. However, apart from that
they seem to differ in several aspects such as: 1. Signature: CQG is Lorentzian
while ASQG is mostly Euclidian. 2. Background Independence (BI): CQG is
manifesly BI while ASQG is apparently not. 3. Truncations: CQG is apparently
free of truncations while ASQG makes heavy use of them.
  The purpose of the present work is to either overcome actual differences or
to explain why apparent differences are actually absent. Thereby we intend to
enhance the contact and communication between the two communities. The focus of
this contribution is on conceptual issues rather than deep technical details
such has high order truncations. On the other hand the paper tries to be
self-contained in order to be useful to researchers from both communities.
  The point of contact is the path integral formulation of Lorentzian CQG in
its reduced phase space formulation which yields the formal generating
functional of physical (i.e. gauge invariant) either Schwinger or Feynman
N-point functions for (relational) observables. The corresponding effective
actions of these generating functionals can then be subjected to the ASQG
Wetterich type flow equations which serve in particular to find the rigorous
generating fuctionals via the inverse Legendre transform of the fixed pointed
effective action.
