By Corollary 5.23 the restriction of the canonical metric of $L^{{\text{\rm an}}}$ is the canonical metric of $\iota_{\sigma}^{\ast}L^{{\text{\rm an}}}$. Therefore, the equality $\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(V(\sigma))=\operatorname{h}^{\operatorname{tor}}_{\iota_{\sigma}^{\ast}{\overline{L}}}(X_{\Sigma(\sigma)})$ follows from Theorem 2.46(2).

To prove the second equality, choose $m_{\sigma}\in F_{\sigma}\cap M$. We will follow the notation of Proposition 5.75. By Theorem 6.6,

$$\operatorname{h}^{\operatorname{tor}}_{\iota_{\sigma}^{\ast}{\overline{L}}}(X_{\Sigma(\sigma)})=(d+1)!\int_{\Delta_{(\Psi-m_{\sigma})(\sigma)}}\vartheta_{\|\cdot\|_{\sigma}}\,\text{\rm d}\operatorname{vol}_{M(\sigma)}.$$

By Proposition 4.47, $\Delta_{(\Psi-m_{\sigma})(\sigma)}=(\pi_{\sigma}^{\vee}+m_{\sigma})^{-1}F_{\sigma}.$ By Proposition 5.75

$$\vartheta_{\|\cdot\|_{\sigma}}=\lambda_{K}\varphi^{\vee}_{\|\cdot\|_{\sigma}}=\lambda_{K}(\pi_{\sigma}^{\vee}+m_{\sigma})^{\ast}\varphi^{\vee}_{\|\cdot\|}=(\pi_{\sigma}^{\vee}+m_{\sigma})^{\ast}\vartheta_{\|\cdot\|}.$$

Since $M(F_{\sigma})=M(\sigma)$, we obtain

$$\int_{\Delta_{(\Psi-m_{\sigma})(\sigma)}}\vartheta_{\|\cdot\|_{\sigma}}\,\text{\rm d}\operatorname{vol}_{M(\sigma)}=\int_{F_{\sigma}}\vartheta_{\|\cdot\|}\,\text{\rm d}\operatorname{vol}_{M(F_{\sigma})},$$

proving the result. ∎