  Given a quotient vector bundle $\mathcal A$ over $X$ with kernel map $\kappa:
X\to\mathrm{Max}\,A$ we study the codual bundle with fiber at each point $x\in
X$ isomorphic to the dual of $\kappa(x)$. 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
$L\to\mathrm{Max}\, A$.
