Consider the single qubit depolarizing noise channel given by
$$\Phi(\rho) = \frac{\lambda}{d} \mathbb{I} + (1- \lambda) \rho.$$
What might be the adjoint $\Phi^{*}(\cdot)$ of this channel? In particular, I am trying to find how the adjoint acts on standard basis states $|0\rangle\langle 0|$ and $|1\rangle\langle 1|$.