Asymptotically Toeplitzness of weighted composition operators

‎‎For a bounded analytic map $\psi$ on the unit disk $\mathbb D$ and analytic self-map $\varphi$ of $\mathbb D$‎, ‎a weighted composition operator $C_{\psi,\varphi}$ on the Hardy space $H^2=H^2(\mathbb D)$ is defined by $C_{\psi,\varphi}f=\psi‎ . ‎f\circ \varphi $‎. ‎Hear we study the asymptotically Toeplitzness of weighted composition operators and their adjoints in different topology on $B(H^2)$‎. ‎For example we show $C_{\psi,\varphi}$ is Toeplitz if and only if it is the Toeplitz operator $T_{\psi}$ and is uniformly asymptotically Toeplitz (UAT) when this operator is compact or Toeplitz‎.