Bimod in nLab

The 1-morphisms are (R,S)(R,S)-bimodules. That is, objects AA in VV with a compatible left action of RR and right action of SS. The composition is given by a generalization of the usual notion of tensor product: For an (R,S)(R,S)-bimodule AA and an (S,T)(S,T)-bimodule BB, a new (R,T)(R,T)-bimodule A⊗ SBA \otimes_S B is computed as the coequaliser of AA‘s right action and BB’s left action of SS.

A⊗S⊗B→ρ⊗B →A⊗λA⊗B→A⊗ SB A \otimes S \otimes B \begin{matrix} \overset{\rho \otimes B}{\rightarrow} \\ \underset{A \otimes \lambda}{\rightarrow} \end{matrix}\,\, A \otimes B \rightarrow A \otimes_S B