definition. equivalence of categories [kostecki2011introduction, 4.3] [tt-001K]

✍️source

The categories \({\cal C}\) and \({\cal D}\) are called equivalent, denoted \({\cal C} \simeq {\cal D}\), iff there exist functors together with natural isomorphisms \[\mathit {1}_{{\cal C}} \cong \mathscr {F} \mathbin {\bullet } \mathscr {G}\] and \[\mathscr {G} \mathbin {\bullet } \mathscr {F} \cong \mathit {1}_{{\cal D}} \]