  We prove an equivalence between cocomplete Yoneda structures and certain
proarrow equipments on a 2-category $\mathcal K$. In order to do this, we
recognize the presheaf construction of a cocomplete Yoneda structure as a
relative, lax idempotent monad sending each admissible 1-cell $f :A \to B$ to
an adjunction $\boldsymbol{P}_!f\dashv\boldsymbol{P}^*f$. Each cocomplete
Yoneda structure on $\mathcal K$ arises in this way from a relative lax
idempotent monad "with enough adjoint 1-cells", whose domain generates the
ideal of admissibles, and the Kleisli category of such a monad equips its
domain with proarrows. We call these structures "yosegi". Quite often, the
presheaf construction associated to a yosegi generates an ambidextrous Yoneda
structure; in such a setting there exists a fully formal version of Isbell
duality.
