Let $\mathbb K$ be a 2-category equipped with a Yoneda structure in the sense of Street-Walters (Street & Walters, 1978) (see also (Street, 1974)), which means $\mathbb K$ is equipped with a right ideal $\mathbb A\subseteq\mathbb K_1$ of “admissible” 1-arrows, and for every $A\in \mathbb A$ a “Yoneda morphism” $Y_A : A \to PA$ is given so that for each $f : A \to B$ in $\mathbb A$ there is a diagram with the property that

• $\chi^f$ exhibits $B(f,1)$ as the left extension of $Y_A$ along $f$;
• $\chi^f$ exhibits $f$ as the absolute lift of $Y_A$ along $B(f,1)$;
• for every $k : B\to PA$ and $\sigma : B(f,1)\To k$, $\sigma$ is an isomorphism in $\mathbb K$ if and only if $(\sigma * f)\circ \chi^f$ exhibits $f$ as absolute lift of $Y_A$ along $k$.

Street and Walters prove that

determines a pseudo-functor $P : \mathbb A \to \mathbb K$, and moreover if $j : A \to B$ is such that $A,B$ and $B(j,1)$ are admissible, then $Pj\dashv\forall_j$, where

Examples of such structure: $\Cat$, and more generally $\VCat$; $\Cat(\mathbb E)$ for a topos $\mathbb E$.

We call uniform Yoneda structure on $\mathbb K$ a Yoneda structure as defined above in the sense of (Street & Walters, 1978), where in addition

• each extension $\chi^f$ is pointwise;
• for every $g : B\to PC$ there exists $Z$ and $B\xto{b} Z \xto{c} C$ such that $g = Pc\circ Y_Z\circ b$.

Examples of such structure: $\Cat$, uniform cosmoi of (Street, 1974).

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.

• In a 2-category $\mathbb K$ with a Yoneda structure, an admissible square $\vf : \esq{S}{U}{T}{V}$ is exact according to BC# if and only if the diagram exhibits $B(U,1)\circ V$ as the left extension of $Y_X\circ S$ along $T$. This is condition PP#
• If the Yoneda structure is uniform conditions BC# and PP# are equivalent for the squares $\vf$ such that $U$ is admissible.
• Relative adjunctions and, more in general, absolute extensions are (as in the case of $\Cat$) exact squares in the sense of BC#.