  We apply categorical machinery to the problem of defining cyclic cohomology
with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf
algebroids. In the case of the former, no definition was thus far available in
the literature, and while a definition exists for the latter, we feel that our
approach demystifies the seemingly arbitrary formulas present there. This paper
emphasizes the importance of working with a biclosed monoidal category in order
to obtain natural coefficients for a cyclic theory that are analogous to the
stable anti-Yetter-Drinfeld contramodules for Hopf algebras.
