Corollary 8.7 Cox

Question

Corollary 8.7 says the following: Let $L,M$ abelian extensions of $K$. Then $L \subseteq M$ $\iff$ there exists a modulus $\mathfrak{m}$ divisible by all primes of $K$ ramified in either $L$ or $M$, such that $$P_{K,1}(\mathfrak{m}) \subseteq \text{ker}(\Phi_{M/K,\mathfrak{m}}) \subseteq \text{ker}(\Phi_{L/K,\mathfrak{m}}).$$ I get the forward part, but the proof of the backward part goes as follows:
Look at the Artin map $\Phi_{M/K,\mathfrak{m}}:I_{K}(\mathfrak{m}) \xrightarrow{} \text{Gal}(L/K)$. Now, the subgroup $\text{ker}(\Phi_{L/K,\mathfrak{m}})$ corresponds to a subgroup under the map, call it $H$. By Galois correspondence, this subgroup corresponds to an intermediate field, say $\overline{L}$. Now, the book says that applying the first part of the proof, we find out that $\text{ker}(\Phi_{\overline{L}/K,\mathfrak{m}}) =\text{ker}(\Phi_{L/K,\mathfrak{m}})$, which I do not understand. I realize there is another question on this forum which has an answer, but that is in terms of ideles which I do not know. Is there any ideal-theoretic solution to this?