definition. (covariant) functor [kostecki2011introduction, 3.1] [tt-0014]

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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.