  This paper examines the category theory of stratified set theory (NF and KF).
We work out the properties of the relevant categories of sets, and introduce a
functorial analogue to Specker's T-operation. Such a development leads one to
consider the appropriate notion of "elementary topos" for stratified set
theories. In addition to considering the categorical properties of a generic
model of NF set theory, we identify a stratified Yoneda Lemma and show NF
encodes itself as a full internal subcategory. Finally, our desire to examine
NF in the context of category theory motivates a more precise examination of
strongly cantorian as an appropriate notion of smallness, replacing it with the
notion of fibrewise strongly cantorian. In the absence of Choice, we introduce
a new axiom (SCU) to NF, and examine some properties of NF + SCU.
