definition. transpose [kostecki2011introduction, 5.1] [tt-001X]

✍️source

Given an adjunction \(\mathscr {L} \dashv \mathscr {R}: {\cal C} \rightleftarrows {\cal D}\), there exists \(f^{\sharp }\) and \(g^{\flat }\) such that the diagrams commute for any arrow \(f: X \to \mathscr {R}(Y)\) in \({\cal C}\), \(g: \mathscr {L}(X) \to Y\) in \({\cal D}\).

\(f^{\sharp }\) is called the left transpose of \(f\). \(g^{\flat }\) is called the right transpose of \(g\).

Other possible terms are left/right adjunct of each other, and mates [nlab2023adjunct].