Proposition [lcc-001S]

The assignment (X,Y,L) \mapsto (X,L^+), (\phi ^+,\phi ^-): L \to L' \mapsto \phi ^+ defines a functor (-)^+: \mathsf {Minmax} \to \mathsf {Conv}

Similarly, (-)^- defines a functor \mathsf {Minmax} \to \mathsf {Conc}^\mathrm {op}. (The reason for this idiosyncratic way of writing a contravariant functor will become apparent in a minute)

The assignment (X,f) \mapsto (X,-f) defines a functor (identity on morphisms) \mathsf {Conc} \to \mathsf {Conv}, and vice versa. Then L^- = -(L^*)^+

The assignment (X,f) \mapsto (X,*, f) defines a fully faithful functor \mathsf {Conv} \to \mathsf {Minmax}, whose essential image consists of those tuples (X,Y,L) where Y is singleton. Analogously, (Y,f) \mapsto (*,Y,f) defines a fully faithful functor \mathsf {Conc}^\mathrm {op} \to \mathsf {Minmax}

We will abuse notation and identify \mathsf {Conv} and \mathsf {Conc} with their images under these inclusions - thus, for example, L^+ will be regarded as an object of \mathsf {Minmax}.

(-)^+ is right adjoint to the inclusion of \mathsf {Conv}, and (-)^- (viewed as a functor \mathsf {Minmax} \to \mathsf {Conc}^\mathrm {op}) is left adjoint to the inclusion of \mathsf {Conc}^\mathrm {op}

Using these identifications, we have (-)^- = (((-)^*)^+)^*