We study the combinatorics of an analogue of Green’s $${\mathcal {J}}$$
J
-relation (a.k.a. the two-sided relation) for the bicategory of finite-dimensional bimodules over finite-dimensional associative algebras over a fixed field. In particular, we provide a number of exameples for both $${\mathcal {J}}$$
J
-equivalent and inequivalent but $${\mathcal {J}}$$
J
-comparabe algebras.