  In this paper, we consider measurings between Hopf algebroids and show that
they induce morphisms on cyclic homology and cyclic cohomology. We also
consider comodule measurings between SAYD modules over Hopf algebroids. These
give an enrichment of the global category of SAYD modules over comodules. These
measurings also induce morphisms on cyclic (co)homology of Hopf algebroids with
SAYD coefficients, which are compatible with Hopf-Galois maps. Finally, we
consider non-$\Sigma$ operads with multiplication. We obtain an enrichment of
cyclic unital comp modules over non-$\Sigma$ operads, as well as morphisms on
cyclic homology induced by measurings of comp modules over operads with
multiplication.
