Let $\mathbb K$ be a 2-category with a uniform Yoneda structure on it.

• A square $\esq{S}{U}{T}{V} : \nbsmat{A &\to&Y\\ \downarrow && \downarrow \\ X & \to&B}$ is called admissible if $S,T,U,X(s,1)$ are admissible, and $Ps$ has a right adjoint $\exists_s$. Then $\vf$ determines $\widetilde{\vf} : \exists_s\circ Y(T,1) \To B(U,1)\circ V$, i.e. a 2-cell filling $\nbsmat{PA &\leftarrow & Y \\ \downarrow && \downarrow \\ PX &\leftarrow & B}$, like in the case of categories and bimodules discussed in §1 (of which this is a generalization).
• If $\vf$ is admissible in the sense above, the square $\esq{S}{U}{T}{V}$ is called exact (according to “condition BC#”) if $\widetilde{\vf}$ is an isomorphism.
• If $\vf$ and $\widetilde{\vf}$ are both admissible, $\vf$ is called $Y$-pointwise if $\widetilde{\vf}$ is also exact.