Measurings of Hopf algebroids and morphisms in cyclic (co)homology theories

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-$\mathrm{\Sigma }$ operads with multiplication. We obtain an enrichment of cyclic unital comp modules over non-$\mathrm{\Sigma }$ operads, as well as morphisms on cyclic homology induced by measurings of comp modules over operads with multiplication.