A tensor-hom adjunction in a topos related to vector topologies and bornologies

In this paper, image is the one-point compactification of the discrete space of natural numbers, M is the monoid of continuous maps f:N→N such that f(∞)=∞, and image is the topos of M-sets. We define two sheaf subtoposes image and image of image and construct a tensor-hom adjunction between certain categories of modules in image and image. Finally, we prove that this construction induces an adjunction between adequate categories of topological and bornological real vector spaces.