definition. identity functor [kostecki2011introduction, 3.1, example 1] [tt-003P]

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The identity functor \(\mathit {1}_{{\cal C}}: {\cal C} \to {\cal C}\) (denoted also by \( 1_{{\cal C}}: {\cal C} \to {\cal C}\)), defined by \(\mathit {1}_{{\cal C}}(X)=X\) and \(\mathit {1}_{{\cal C}}(f)=f\) for every \(X \in \operatorname {Ob}({\cal C})\) and every \(f \in \operatorname {Arr}({\cal C})\).