Trying to work out the CP-transformation property of the Higgs potential

The parity transformation property of a complex scalar field $phi(x)$ is given by: $$Pphi(t,textbf{x}) P^{-1}=eta_Pphi(t,-textbf{x})$$ where $eta_P=pm 1$. The charge conjugation property of a complex scalar field $phi(x)$ is given by: $$Cphi(t,textbf{x}) C^{-1}=eta_Cphi^dagger(t,textbf{x})$$ where $eta_C=pm 1$. Therefore, the CP transformation property of $phi(x)$ can be worked out to be $$(CP)phi(x)(CP)^{-1}=C(Pphi(t,textbf{x}) P^{-1})C^{-1}=eta_PCphi (t,-textbf{x})C^{-1}=eta_Peta_Cphi^dagger(t,-textbf{x})$$
$$Rightarrow (CP)phi(x)(CP)^{-1}=eta_{CP}phi^dagger(t,-textbf{x})tag{1}$$ where $eta_{CP}=eta_Peta_C=pm 1$.

How will the CP-transformation property change if $H(x)$ is a SU(2) doublet, such as the Higgs field of the standard model $H(x)=begin{pmatrix}phi_1(x)& phi_{2}(x)end{pmatrix}^T$? Can I directly use (1) for the doublet $H(x)$ itself? If yes, how do we work out the action of CP on the doublet $H(x)$? Is it like $$(CP)H(CP)^{-1}=begin{pmatrix}(CP)phi_1(x)(CP)^{-1} (CP)phi_{2}(x)(CP)^{-1}end{pmatrix}=pm begin{pmatrix}phi^{dagger}_1(t,-textbf{x}) phi_{2}^dagger(t,-textbf{x})end{pmatrix}=pm H^{dagger}(t,-textbf{x})?tag{2}$$ To be concrete, I want to check the CP-transformation property of the Higgs potential of the Standard model given by $$V(H)=mu^2(H^dagger H)+lambda(H^dagger H)^2.$$