  The reflexive completion of a category consists of the Set-valued functors on
it that are canonically isomorphic to their double conjugate. After reviewing
both this construction and Isbell conjugacy itself, we give new examples and
revisit Isbell's main results from 1960 in a modern categorical context. We
establish the sense in which reflexive completion is functorial, and find
conditions under which two categories have equivalent reflexive completions. We
describe the relationship between the reflexive and Cauchy completions,
determine exactly which limits and colimits exist in an arbitrary reflexive
completion, and make precise the sense in which the reflexive completion of a
category is the intersection of the categories of covariant and contravariant
functors on it.
