The Morita bicategory of rings and bimodules

The Morita bicategory of rings and bimodules $\mathbf{B}\mathbf{i}\mathbf{m}\mathbf{o}\mathbf{d}$ has as objects the unital rings. For unital rings $A,B$ the hom-category $\mathbf{B}\mathbf{i}\mathbf{m}\mathbf{o}\mathbf{d}(A,B)$ is the category whose objects are the $(B,A)$-bimodules and whose morphisms $f:M\to M^{\mathrm{\prime }}$ are the maps that are simultaneously left $B$-linear and right $A$-linear, with identity maps as identities and composition of functions as composition. The identity 1-cell of $A$ is the regular bimodule ${}_A{A}_A$. Composition of a $(C,B)$-bimodule $N$ with a $(B,A)$-bimodule $M$ is the tensor product $N{\otimes }_{B}M$, a $(C,A)$-bimodule by A commuting outer scalar action descends to a tensor product; on maps it is $(g,f)\mapsto g\otimes f$ (Module homomorphisms induce tensor-product homomorphisms functorially). The associator and the unitors are the canonical isomorphisms of Associativity of tensor products for compatible bimodules and The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$. These data form a bicategory in the sense of Bicategories, pseudofunctors, and biequivalences; the coherence check is The Morita data satisfy the bicategory coherence axioms ↗. The handedness is the one fixed for this expansion: modules are left modules, so that a 1-cell $A\to B$ is a $(B,A)$-bimodule and its tensor functor is $T_M(X)=M{\otimes }_{A}X$, and a $(C,B)$-bimodule $N$ composes with $M$ as $N{\otimes }_{B}M$. No commutativity of the rings is assumed and no choice is used.