What is Isbell completion?

Dear All, Let A be a small category. Isbell conjugacy gives an adjunction between Set^(A^op) and (Set^A)^op. Like any adjunction, this restricts in a maximal way to an equivalence between full subcategories. Write I(A) for either side of that equivalence. What is I(A)? Certainly I(A) contains the Cauchy-completion of A. But it can be strictly bigger: for example, if A is the initial category then I(A) is the terminal category. Or, if A is a discrete category with more than one object then I(A) is A with initial and terminal objects adjoined. One can also ask the question in an enriched setting. Best wishes, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]