  New Foundations ($\mathrm{NF}$) is a set theory obtained from naive set
theory by putting a stratification constraint on the comprehension schema; for
example, it proves that there is a universal set $V$. $\mathrm{NFU}$
($\mathrm{NF}$ with atoms) is known to be consistent through its close
connection with models of conventional set theory that admit automorphisms.
  A first-order theory, $\mathrm{ML}_\mathrm{CAT}$, in the language of
categories is introduced and proved to be equiconsistent to $\mathrm{NF}$
(analogous results are obtained for intuitionistic and classical $\mathrm{NF}$
with and without atoms). $\mathrm{ML}_\mathrm{CAT}$ is intended to capture the
categorical content of the predicative class theory of $\mathrm{NF}$.
$\mathrm{NF}$ is interpreted in $\mathrm{ML}_\mathrm{CAT}$ through the
categorical semantics. Thus, the result enables application of category
theoretic techniques to meta-mathematical problems about $\mathrm{NF}$ -style
set theory. For example, an immediate corollary is that $\mathrm{NF}$ is
equiconsistent to $\mathrm{NFU} + |V| = |\mathcal{P}(V)|$. This is already
proved by Crabb\'e, but becomes more transparent in light of the results of
this paper.
  Just like a category of classes has a distinguished subcategory of small
morphisms, a category modelling $\mathrm{ML}_\mathrm{CAT}$ has a distinguished
subcategory of type-level morphisms. This corresponds to the distinction
between sets and proper classes in $\mathrm{NF}$. With this in place, the axiom
of power objects familiar from topos theory can be appropriately formulated for
$\mathrm{NF}$. It turns out that the subcategory of type-level morphisms
contains a topos as a natural subcategory.
