Isbell conjugacy

Isbell conjugacy (named after John R. Isbell) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986.
Definition.
Let formula_1 be a symmetric monoidal closed category, and let formula_2 be a small category enriched in formula_1.
The Isbell conjugacy is an adjunction between the categories formula_4 and formula_5 arising from the Yoneda embedding formula_6 and the dual Yoneda embedding formula_7.