definition. opposite category [kostecki2011introduction, 2.5] [tt-000H]
✍️source

A category is called the opposite category of \({\cal C}\), denoted \(C^{op}\), iff

(reversion of arrows) \[\operatorname {Ob}({\cal C}^{op}) = \operatorname {Ob}({\cal C})\] \[\operatorname {Arr}({\cal C}^{op}) \ni f: Y \to X \Longleftrightarrow \operatorname {Arr}({\cal C}) \ni f: X \to Y\]
(reversion of composition)

special objects and categories [tt-000G]