  This paper is primarily concerned with assessing a set-theoretical system,
$S^*$, for the foundations of category theory suggested by Solomon Feferman.
$S^*$ is an extension of NFU, and may be seen as an attempt to accommodate
unrestricted categories such as the category of all groups (without any
small/large restrictions), while still obtaining the benefits of ZFC on part of
the domain. A substantial part of the paper is devoted to establishing an
improved upper bound on the consistency strength of $S^*$. The assessment of
$S^*$ as a foundation of category theory is framed by the following general
desiderata (R) and (S). (R) asks for the unrestricted existence of the category
of all groups, the category of all categories, the category of all functors
between two categories, etc., along with natural implementability of ordinary
mathematics and category theory. (S) asks for a certain relative distinction
between large and small sets, and the requirement that they both enjoy the full
benefits of the $\mathrm{ZFC}$ axioms. $S^*$ satisfies (R) simply because it is
an extension of NFU. By means of a recursive construction utilizing the notion
of strongly cantorian sets, we argue that it also satisfies (S). Moreover, this
construction yields a lower bound on the consistency strength of $S^*$. We also
exhibit a basic positive result for category theory internal to NFU that
provides motivation for studying NFU-based foundations of category theory.
