  We develop category theory within Univalent Foundations, which is a
foundational system for mathematics based on a homotopical interpretation of
dependent type theory. In this system, we propose a definition of "category"
for which equality and equivalence of categories agree. Such categories satisfy
a version of the Univalence Axiom, saying that the type of isomorphisms between
any two objects is equivalent to the identity type between these objects; we
call them "saturated" or "univalent" categories. Moreover, we show that any
category is weakly equivalent to a univalent one in a universal way. In
homotopical and higher-categorical semantics, this construction corresponds to
a truncated version of the Rezk completion for Segal spaces, and also to the
stack completion of a prestack.
