Conjugacy class APS

Definition
Let $$(G,\Phi)$$ be an APS of groups. The conjugacy class APS of $$G$$ is an APS of sets whose $$n^{th}$$ member is $$C(G_n)$$, the set of conjugacy classes of $$G_n$$, and where the concatenation map from $$C(G_m) \times C(G_n)$$ is the same as $$\Phi_{m,n}$$, upto conjugacy.

More sophisticatedly, the equivalence relation of being conjugate is compatible with the concatenation, and the quotient of the original APS by this equivalence relation gives the conjugacy class APS.