One of the generalizations of algebraic geometry is provided by the theory of semiring schemes, cf. Lorscheid 2012. The theory follows the same set up of scheme theory, but we use semirings instead of rings.

Given a semiring $R$, we have a semiringed space $\mathrm{Spec}(R)$, defined by mimicking the usual definition for rings. This gives a functor $\mathrm{Spec}$ from the opposite of the category semirings to that of affine semiring schemes.

Conversely, there's also a global sections functor $\Gamma:\mathrm{AffSemiSch}^\circ\to\mathrm{Semiring}$ sending a semiringed space $(X,\mathcal{O}_X)$ to $\Gamma(X,\mathcal{O}_X)$.

Does the pair $(\mathrm{Spec},\Gamma)$ give as in ordinary algebraic geometry a contravariant equivalence of categories $\mathrm{Semiring}\cong\mathrm{AffSemiSch}^\circ$?