  The Grothendieck universe axiom asserts that every set is a member of some
set-theoretic universe U that is itself a set. One can then work with entities
like the category of all U-sets or even the category of all locally U-small
categories, where U is an "arbitrary but fixed" universe, all without worrying
about which set-theoretic operations one may legitimately apply to these
entities. Unfortunately, as soon as one allows the possibility of changing U,
one also has to face the fact that universal constructions such as limits or
adjoints or Kan extensions could, in principle, depend on the parameter U. We
will prove this is not the case for adjoints of accessible functors between
locally presentable categories (and hence, limits and Kan extensions), making
explicit the idea that "bounded" constructions do not depend on the choice of
U.
