The left adjoint of Spec from a category of lattice-ordered groups

Let us write $\ell G_u^f$ for the category whose objects are lattice-ordered abelian groups (l-groups for short) with a strong unit and finite prime spectrum endowed with a collection of Archimedean elements, one for each prime l-ideal, which satisfy certain properties, and whose arrows are l-homomorphisms with additional structure. In this paper we show that a functor which assigns to each object $(A,\hat{u})\in \ell G_u^f$ the prime spectrum of A, and to each arrow $f:(A,\hat{u})\to (B,\hat{v})\in \ell G_u^f$ the naturally induced p-morphism, has a left adjoint.