Street and Walters prove that
$$
PB(Y_B\circ f,1) =: Pf
$$
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
$$
PA(B(j,1),1)=:\forall_j
$$
Examples of such structure: $\Cat$, and more generally $\VCat$; $\Cat(\mathbb E)$ for a topos $\mathbb E$.