If we take the Morita bicategory which has k-algbras as objects, bimodules as 1-morphisms and bimodule maps as 2-morphisms, then what would the unitors be explicitly? For algebra $A$, I can take the evaluation and coevaluation 1-morphisms to be $_{A \otimes_k A^{op}}A_k$ and $_kA_{A^{op} \otimes_k A}$ respectively. So would the unitors be $_AA_{A\otimes_k k }$ and $_{A\otimes_k k }A_A$ respectively?
If so, then what should the explicit construction of the cusp isomorphisms, $\alpha$ and $\beta$ in following picture, be?
digram of required isomorphism
Ignoring the unitors and associators in the diagram, one answer I have is the following: $$\alpha : (E \otimes id) \circ (id \otimes C) \rightarrow id$$ $$ (_AA_A \otimes_{k} \ _kA_{A^{op} \otimes_k A}) \otimes (_{A \otimes_k A^{op}}A_k \otimes_k \ _AA_A) \rightarrow _AA_A$$ $$(a_1 \otimes a_2) \otimes (a_3 \otimes a_4) \mapsto a_1a_3a_2a_4$$
However there is the issue of what the $\alpha^{-1}$ would be in this situation. Since each permutation of the 4 factors would map to something different under $\alpha^{-1}$, how do I show that it is all the same, i.e, well defined no matter if I start with $a = a_1a_2a_3a_4$ or $a = a_4a_2a_1a_3$ and apply $\alpha^{-1}$? How should I define $\alpha^{-1}$ explicitly?