The codual quotient vector bundle

Given a quotient vector bundle $\mathcal{A}$ over $X$ with kernel map $\kappa: X\to$\mathrm{M}\mathrm{a}\mathrm{x}$,A$westudythecodualbundlewithfiberateachpoint$x\in X$isomorphictothedualof$\kappa(x)$. Applying the adjunction between quotient vector bundles and linearized locales to the codual bundle leads to a new adjunction between the category of all quotient vector bundles and a category whose objects are locales $L$ equipped with a contravariant morphism $L\to \mathrm{M}\mathrm{a}\mathrm{x}A$.