For a small category , just as the presheaf category is the free cocompletion of , by duality we can conclude that is the free completion of . This means that any functor where is complete factors uniquely (up to isomorphism) through the “dual Yoneda embedding” via a continuous functor .
This operation is a 2-monad which is colax idempotent but not (even weakly) idempotent. In analogy with the general concept of completion, we might call the operation of any colax idempotent monad on a -category a ‘free completion’. See discussion at completion.