 Duality of copositive and completely positive cones

Under the nonnegative-pairing convention for the dual cone, ${\mathrm{CP}}_n^{\ast }={\mathrm{COP}}_n$ follows from $⟨A,x{x}^T{⟩}_F=x^{T}Ax$. Conversely, if $B\notin {\mathrm{CP}}_n$, separation from a closed convex cone gives a separating $A$ nonnegative on all generators and negative on $B$. That $A$ is a copositive matrix, so $B\notin {\mathrm{COP}}_n^{\ast }$. This proves the displayed equality using closedness of the completely positive cone.