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Faltings initiated in 2005 a p-adic analogue of the (complex) Simpsoncorrespondence whose construction has been taken up by various authors,according to several approaches. Following the one we initiated previously, wedevelop in this new monograph new features of the p-adic Simpsoncorrespondence, inspired by our construction of the relative Hodge-Tatespectral sequence. First, we address the connection to Hodge-Tate localsystems. Second, we establish the functoriality of the p-adic Simpsoncorrespondence by proper direct image. Along the way, we expand the scope ofour original construction.  Faltings a d\'egag\'e en 2005 un analogue p-adique de la correspondance deSimpson (complexe) dont la construction a \'et\'e reprise par diff\'erentsauteurs, selon plusieurs approches. Poursuivant celle que nous avons initi\'eepr\'ec\'edemment, nous d\'eveloppons dans la pr\'esente monographie de nouveauxaspects de la correspondance de Simpson p-adique, inspir\'es par notreconstruction de la suite spectrale de Hodge-Tate relative. Nous traitons toutd'abord du lien avec les syst\`emes locaux de Hodge-Tate. Nous \'etablissonsensuite la fonctorialit\'e de la correspondance de Simpson p-adique par imagedirecte propre. Chemin faisant, nous \'elargissons la port\'ee de notreconstruction initiale.

Correspondance de Simpson p-adique II : fonctorialit\'e par image directe propre et syst\`emes locaux de Hodge-Tate