  We introduce a categorical approach to classifying actions of C$^*$-tensor
categories $\mathcal{C}$ on C$^*$-algebras up to cocycle conjugacy. We show
that, in this category, inductive limits exist and there is a natural notion of
approximate unitary equivalence. Then, we generalise classical Elliott
intertwining results to the $\mathcal{C}$-equivariant case, in the same fashion
as done by Szab\'o for the group equivariant case in [39].
