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 C ϕ ⁎ C ϕ is strongly asymptotically Toeplitz for all analytic self-maps ϕ of the unit disk, and that C ϕ C ϕ ⁎ is Toeplitz if and only if ϕ is the identity or a rotation. Also, we see that C ϕ C ϕ ⁎ can exhibit varying degrees of asymptotic Toeplitzness.