  We study the biclosedness of the monoidal categories of modules and comodules
over a (left or right) Hopf algebroid, along with the bimodule category centres
of the respective opposite categories and a corresponding categorical
equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd
modules, respectively. This is directly connected to the existence of a trace
functor on the monoidal categories of modules and comodules in question, which
in turn allows to recover (or define) cyclic operators enabling cyclic
cohomology.
