definition. contravariant functor [kostecki2011introduction, 3.1] [tt-0015]

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A functor \(\mathscr {F}\) is called a contravariant functor from \({\cal C}\) to \({\cal D}\), and denoted \(\mathscr {F} : {\cal C}^{op} \to {\cal D}\), if it obeys the definition given by the (covariant) functor for \({\cal C}\) replaced by \({\cal C}^{op}\), i.e. it's given by the diagram i.e. a map of objects and arrows between categories \({\cal C}\) and \({\cal D}\) that reverses the structure of the arrows, compositions and identities.