I am stuck on a small detail in the proof of Proposition 10.4(iii) from Chapter VII of Neukirch's Algebraic Number Theory.
For a Galois extension of number fields $L|K$ and a representation $(\rho, V)$ of $\text{Gal}(L|K)$ with character $\chi$, define the Artin $L$-series $$\mathcal{L}(L|K, \chi, s)=\prod_{\mathfrak{p}} \frac{1}{\det(1-\rho(\varphi_{\mathfrak P})N(\mathfrak p)^{-s}; V^{I_{\mathfrak{P}}})}$$ where $\varphi_{\mathfrak P}$ is the Frobenius element, i.e. the generator of $G_{\mathfrak P}/I_{\mathfrak P}$, where $\mathfrak P$ is a prime in $L$ lying above a prime $\mathfrak p$ in $K$. Here $G_{\mathfrak P} = \{\sigma \in \text{Gal}(L|K)\mid \sigma(\mathfrak P)=\mathfrak P\}$ is the decomposition group and $I_{\mathfrak P}=\{\sigma \in \text{Gal}(L|K)\mid \sigma (x)\equiv x \pmod{\mathfrak P}\}$ is the inertia group.
Now the Proposition I am referring to is the following:
Proposition. For a bigger Galois extension $L'\supseteq K$ such that $L'\supseteq L\supseteq K$ and a character $\chi$ of $G=\text{Gal}(L|K)$ we have $\mathcal{L}(L'|K, \chi, s)=\mathcal{L}(L|K, \chi, s).$
Proof. Let $\mathfrak{P}'|\mathfrak{P}|\mathfrak{p}$ be primes of $L'|L|K$, each lying above the next. Let $\chi$ be a character belonging to the $\mathbb{C}G$-module $V$. Now $\text{Gal}(L'|K)$ acts on $V$ via the projection $\text{Gal}(L'|K)\to \text{Gal}(L|K)$. This projection induces surjective homomorphisms $$G_{\mathfrak{P}'}\to G_{\mathfrak{P}}, I_{\mathfrak{P}'}\to I_{\mathfrak{P}}, G_{\mathfrak{P}'}/I_{\mathfrak{P}'}\to G_{\mathfrak{P}}/I_{\mathfrak{P}}.$$ The latter maps the Frobenius automorphism $\varphi_{\mathfrak{P}'}$ to the Frobenius automorphism $\varphi_{\mathfrak P}$ so that $(\varphi_{\mathfrak{P}'}, V^{I_{\mathfrak{P}'}})=(\varphi_{\mathfrak{P}}, V^{I_{\mathfrak{P}}})$ i.e. $$\det(1-\rho(\varphi_{\mathfrak{P}'})t; V^{I_{\mathfrak{P}'}})=\det(1-\rho(\varphi_{\mathfrak{P}})t; V^{I_{\mathfrak{P}}}).$$
Now the main part of the proof I don't understand is where they claim that $(\varphi_{\mathfrak{P}'}, V^{I_{\mathfrak{P}'}})=(\varphi_{\mathfrak{P}}, V^{I_{\mathfrak{P}}})$. What does this really mean? It can't be an equality of pairs since $\varphi_{\mathfrak{P}'}$ isn't necessarily the same as $\varphi_{\mathfrak{P}}$. It also wouldn't be clear to me why we would have $V^{I_{\mathfrak{P}'}}=V^{I_{\mathfrak P}}$.
I hope my confusion is clear and I can try to explain more if needed. Thank you in advance!