The section $s^{\prime}$ is a nowhere vanishing section over $X_{\Sigma,\sigma}$. Therefore, the function $g_{{\overline{L}},s^{\prime}}\colon X^{{\text{\rm an}}}_{\Sigma,\sigma}\to\mathbb{R}$ of diagram (5.13) can be extended to a continuous function on $X_{\Sigma,\sigma}$ that we also denote $g_{{\overline{L}},s^{\prime}}$. By the definition of the inverse image of a metric, there is a commutative diagram

Then the result is a consequence of the definition of $\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}$ and of the commutativity of the diagram

that follows from Proposition 5.9. ∎