A completeness conjecture is advanced concerning the free small-colimit
completion P(A) of a (possibly large) category A. The conjecture is based on
the existence of a small generating-cogenerating set of objects in A. We sketch
how the validity of the result would lead to the existence of an Isbell-Lambek
bicompletion C(A) of such an A, without a "change-of-universe" procedure being
necessary to describe or discuss the bicompletion.