  We provide the rigorous foundations for a categorical approach to the
classification of C*-dynamics up to cocycle conjugacy. Given a locally compact
group $G$, we consider a category of (twisted) $G$-C*-algebras, where morphisms
between two objects are allowed to be equivariant maps or exterior
equivalences, which leads to the concept of so-called cocycle morphisms. An
isomorphism in this category is precisely a cocycle conjugacy in the known
sense. We show that this category allows sequential inductive limits, and that
some known functors on the usual category of $G$-C*-algebras extend. After
observing that this setup allows a natural notion of (approximate) unitary
equivalence, the main aim of the paper is to generalize the fundamental
intertwining results commonly employed in the Elliott program for classifying
C*-algebras. This reduces a given classification problem for C*-dynamics to the
prevalence of certain uniqueness and existence theorems, and may provide a
useful alternative to the Evans--Kishimoto intertwining argument in future
research.
