  We want to replace categories, functors and natural transformations by
categories, open functors and open natural transformations. In analogy with
open dynamical systems, the adjective open is added here to mean that some
external information is taken into account. For the particular use of the
authors, such an open functor is described by two components: a presheaf
representing the possible external influences for each input, and a classical
functor from the category of elements of this presheaf to the category of
results. Considering the appropriate notion of composition then leads to a
bicategory. This report describes this bicategory with as little auxiliary
constructions as possible and gives all the details of all the proofs needed to
establish the bicategory, as explicitly as possible. Subsequent reports will
give other presentations of this bicategory and compare it to other existing
constructions, e.g. spans, fibrations, pseudoadjunctions, Kleisli bicategories
of pseudo-monads, and profunctors (or distributors).
