The Grothendieck construction gives an equivalence between fibred categories and indexed categories. This paper gives two monoidal variants of each of these concepts, and the corresponding equivalence. A connection between the two variants is also given.
Mitchell Buckley?, Fibred 2-categories and bicategories, Journal of Pure and Applied Algebra 218(6), pages 1034-1074, 2014. arXiv:1212.6283, DOI
Brian Day? and Ross Street?, Monoidal bicategories and Hopf algebroids, Adv. Math., 129(1):99–157, 1997.