Toeplitzness of products of composition operators and their adjoints

We examine the Toeplitzness of products of composition operators and their adjoints. We show, among other things, that Cϕ⁎Cϕ is strongly asymptotically Toeplitz for all analytic self-maps ϕ   of the unit disk, and that CϕCϕ⁎ is Toeplitz if and only if ϕ   is the identity or a rotation. Also, we see that CϕCϕ⁎ can exhibit varying degrees of asymptotic Toeplitzness.