Let $\psi$ be a continuous metric on $K_{X}^{\mathrm{an}}$, $\psi^{\prime}$ the metric on $K_{X^{\prime}}^{\mathrm{an}}\simeq p^{*}K_{X}^{\mathrm{an}}$ corresponding to $p^{*}\psi$, and set $\kappa^{\prime}:=A_{X^{\prime}}-\psi^{\prime}$. Then $\kappa^{\prime}=mp^{*}\kappa$. As a consequence, $\operatorname{Sk}(\psi^{\prime})=p^{-1}\operatorname{Sk}(\psi)$ and $\kappa^{\prime}_{\min}=m\kappa_{\min}$.