  Temporal logics stands for a widely adopted family of formalisms for the
verification of computational devices, enriching propositional logics by
operators predicating on the step-wise behaviour of a system. Its quantified
extensions allow to reason on the properties of the individual components of
the system at hand. The expressiveness of the resulting logics poses problems
in correctly identifying a semantics that exploit its features without
resorting to the imposition of restrictions on the acceptable behaviours. In
this paper we address this issue by means of counterpart models and relational
presheaves.
