Formal similarities between an antipode for a bialgebra and a dualization functor for a monoidal category were clarified in the article of \textit{B. Day, P. McCrudden} and \textit{R. Street} [``Dualizations and antipodes'', Appl. Categ. Struct. 11, No.~3, 229--260 (2003; Zbl 1137.18302)] via the notion of autonomous pseudomonoid in a monoidal bicategory. The author's goal in this and subsequent papers is to generalize various theorems on Hopf algebras to autonomous pseudomonoids and so to obtain other applications. This paper concentrates on the structure theorem for Hopf modules proved in \textit{R. G. Larson} and \textit{M. E. Sweedler} [``An associative orthogonal bilinear form for Hopf algebras'', Am. J. Math. 91, 75--94 (1969; Zbl 0179.05803)] and the quasi-Hopf algebra version in \textit{P. Schauenburg} [``Two characterizations of finite quasi-Hopf algebras'', J. Algebra 273, No.~2, 538--550 (2004; Zbl 1041.16032)]. More precisely, the author shows that a map pseudomonoid has a left dualization if and only if it satisfies an appropriate version of the fundamental theorem of Hopf modules. He also shows great skill at working in a monoidal bicategory.