definition. isomorphism of categories [kostecki2011introduction, 4.3] [tt-001J]

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The cateories \({\cal C}\) and \({\cal D}\) are called isomorphic, denoted \({\cal C} \cong {\cal D}\), iff there exists functors such that \[\mathit {1}_{{\cal C}}=\mathscr {F} \mathbin {\bullet } \mathscr {G}\] and \[\mathit {1}_{{\cal D}}=\mathscr {G} \mathbin {\bullet } \mathscr {F} \]