definition. constant functor [kostecki2011introduction, 3.1, example 2] [tt-003Q]

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The constant functor \(\Delta _O: {\cal C} \to {\cal D}\) which assigns a fixed \(O \in \mathrm {Ob}({\cal D})\) to any object of \({\cal C}\) and \(\mathit {1}_O\), the identity arrow on \(O\), to any arrows from \({\cal C}\) : with compositions and identities preserved in a trivial way.