Module Eilenberg-Watts calculus

The categorical formulation of the Eilenberg-Watts calculus relates, for any pair finite categories M and N, Fun^{le}(N,M) Fun^{re}(N,M) linear left or right exact functors Deligne product \bar N \boxtimes by adjoint equivalences. We establish a variant this case that are module over tensor category. This provides in particular canonical explicitly computable equivalences between (or, more generally, balanced functors) certain twisted centers bimodule categories.