My question is a more detailed version of the last paragraph in this previous one that I asked.
Given a ring $k$, in the bicategory of $k$-algebras, bimodules (as the 1-morphisms) and intertwiners (2-morphisms), how exactly do the modules act on the algebras? In the sense that if we take the bimodule ${}_AA_A$ which is the identity here, then ${}_AA_A : A \rightarrow A$ means $a \mapsto a ,\ \forall a \in A $. How then do all the other 1-morphisms act on the algberas explicitly?
evaluation is given by ${}_{A^e}A_k : A \otimes A^{op} \rightarrow k$ would mean $ a \otimes b \mapsto \lambda_?$. Similarly coevaluation is given by ${}_kA_{^eA} : k \rightarrow A^{op} \otimes A$ would mean $ \lambda \mapsto b_? \otimes a_?$. Now I want to know what these $\lambda_? , a_?, b_?$ would be as it would explain explicitly how these modules are maps.
So subject to the zigzag equation I would get that $a \otimes \lambda \mapsto a \otimes b_? \otimes a_? \mapsto ev(a \otimes b_?) \otimes a_?$ and another similar equation. How exactly from these two equations can I pinpoint what the possible "values" are?
My motivation for this question is that if I end up with two two 1-morphisms (each of which are composites of different 1-morphisms) which are 2-isomorphic, then I should be able to figure out what that 2-ismorphism is specifically by getting a hold of all the underlying data explicitly. So in this bicategory I would end up with two bimodules each of which is some composition of various bimodules and then try to figure out what the isomorphism (intertwiner) is. How do I go about this? This is also particularly confusing in say the case of ${}_AA_A$ and ${}_{A^e}A_A$ are isomorphic but as identity and evaluation maps their source and targets are completely different.