Parabolic subgroups inside parabolic subgroups of Artin groups

We prove that a parabolic subgroup P P contained in another P prime"> ′ encoding="application/x-tex">P’ of an Artin group A"> A encoding="application/x-tex">A is . This answers question Godelle which not obvious despite appearances. In order to achieve our result we construct set-retraction A right-arrow upper → encoding="application/x-tex">A \to P the inclusion map from into retraction was implicitly constructed previous paper by Charney and second author [Bull. Lond. Math. Soc. 46 (2014), pp. 1248–1255].