definition. adjoint functor [kostecki2011introduction, 5.1] [tt-001Q]

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Given functors we say \(\mathscr {L}\) and \(\mathscr {R}\) are a pair of adjoint functors, or together called an adjunction between them, \(\mathscr {L}\) is called left adjoint to \(\mathscr {R}\), and \(\mathscr {R}\) is called right adjoint to \(\mathscr {L}\), denoted \[\mathscr {L} \dashv \mathscr {R} : {\cal C} \rightleftarrows {\cal D}\] or iff there exists a natural isomorphism \(\sigma \) between the following two hom-bifunctors: \[ {\cal D}(\mathscr {L}(-), =) \cong {\cal C}(-, \mathscr {R}(=)) \] diagramatically,

The components of the natural isomorphism \(\sigma \) are isomorphisms \[ \sigma _{XY} : {\cal D}(\mathscr {L}(X), Y) \cong {\cal C}(X, \mathscr {R}(Y)) \]