  In this paper we introduce the categorical "true local terms" maps for Artin
stacks and show that they are additive and commute with proper pushforwards,
smooth pullbacks and specializations. In particular, we generalizing results of
[Va2] to this setting.
  As an application, we supply proofs of two theorems stated in [AGKRRV].
Namely, we show that the "true local terms" of the Frobenius endomorphism
coincide with the "naive local terms" and that the "naive local terms" commute
with !-pushforwards. The latter result is a categorical version of the
classical Grothendieck--Lefschetz trace formula.
