Definition. (covariant) functor [kostecki2011introduction, 3.1] [tt-0014]
✍️source

    A (covariant) functor \(\mathscr {F}: {\cal C} \to {\cal D}\) is given by the diagram

    i.e. a map of objects and arrows between categories \({\cal C}\) and \({\cal D}\) that preserves the structure of the compositions and identities.