definition. representable functor [kostecki2011introduction, 4.4] [tt-001O]

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A set-valued functor \(\mathscr {F}: {\cal C} \to \mathbf {Set}\) is called covariantly representable if for some \(X \in {\cal C}\), \[\tau : \mathscr {F} \cong {\cal C}(X,-)\] where \(\cong \) denotes a natural isomorphism.

Conversely, a set-valued functor \(\mathscr {G} : {\cal C}^{op} \to \mathbf {Set}\) is called contravariantly representable if for some \(X \in {\cal C}\), \[\tau : \mathscr {G} \cong {\cal C}(-, X)\]

Such an object \(X\) is called a representing object for the functor \(\mathscr {F}\) or \(\mathscr {G}\), respectively.

The pair \((\tau , X)\) is called a representation of the functor \(\mathscr {F}\) (respectively, \(\mathscr {G}\) ).