definition. constant object functor [leinster2016basic, 4.1.6] [tt-003H]

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A functor from the terminal category \(\mathbf {1}\) to a category \({\cal C}\) simply picks out an object of \({\cal C}\), called a constant object functor (which is a constant functor), denoted \(\Delta _X : \mathbf {1} \to {\cal C}\) for some \(X \in \operatorname {Ob}({\cal C})\), or simly denoted by the object, e.g. \(X\).

As special cases, constant object functor for initial and terminal objects are denoted by \(\mathrm {0}\) and \(\mathrm {1}\), respectively.