  A problem is a multivalued function from a set of \emph{instances} to a set
of \emph{solutions}. We consider only instances and solutions coded by sets of
integers. A problem admits preservation of some computability-theoretic
weakness property if every computable instance of the problem admits a solution
relative to which the property holds. For example, cone avoidance is the
ability, given a non-computable set $A$ and a computable instance of a problem
$\mathsf{P}$, to find a solution relative to which $A$ is still non-computable.
  In this article, we compare relativized versions of computability-theoretic
notions of preservation which have been studied in reverse mathematics, and
prove that the ones which were not already separated by natural statements in
the literature actually coincide. In particular, we prove that it is equivalent
to admit avoidance of 1 cone, of $\omega$ cones, of 1 hyperimmunity or of 1
non-$\Sigma^0_1$ definition. We also prove that the hierarchies of preservation
of hyperimmunity and non-$\Sigma^0_1$ definitions coincide. On the other hand,
none of these notions coincide in a non-relativized setting.
