  We prove a relative Lefschetz-Verdier theorem for locally acyclic objects
over a Noetherian base scheme. This is done by studying duals and traces in the
symmetric monoidal $2$-category of cohomological correspondences. We show that
local acyclicity is equivalent to dualizability and deduce that duality
preserves local acyclicity. As another application of the category of
cohomological correspondences, we show that the nearby cycle functor over a
Henselian valuation ring preserves duals, generalizing a theorem of Gabber.
