Comma 2-comonad I: Eilenberg-Moore 2-category of colax coalgebras

In this paper we describe a comma 2-comonad on the 2-category whose objects are functors, 1-cell are colax squares and 2-cells are their transformations. We give a complete description of the Eilenberg-Moore 2-category of colax coalgebras, colax morphisms between them and their transformations and we show how many fundamental constructions in formal category theory like adjoint triples, distributive laws, comprehension structures, Frobenius functors etc. naturally fit in this context.