  In two papers we noted that in common practice many algebraic constructions
are defined only `up to isomorphism' rather than explicitly. We mentioned some
questions raised by this fact, and we gave some partial answers. The present
paper provides much fuller answers, though some questions remain open. Our main
result says that there is a transitive model of Zermelo-Fraenkel set theory
with choice (ZFC) in which every fully definable construction is `weakly
natural' (a weakening of the notion of a natural transformation). A corollary
is that there are models of ZFC in which some well-known constructions, such as
algebraic closure of fields, are not explicitly definable. We also show that
there is no model of ZFC in which the explicitly definable constructions are
precisely the natural ones.
