Linearly distributive categories (LDCs), introduced by Cockett and Seely to model multiplicative linear logic, are categories equipped with two monoidal structures that interact via linear distributivities. The aim of this work was to characterize a particular class of LDCs: the cartesian linearly distributive categories. A key result in monoidal category theory is the Fox theorem, which characterizes cartesian categories. To extend the Fox theorem to LDCs, medial linearly distributive categories, medial linear functors, and medial linear transformations were introduced. Medial LDCs are the categorical structure at the intersection of LDCs and duoidal categories. A LD-Fox adjunction between the inclusion of the 2-category of cartesian LDCs in the 2-category of symmetric strong medial LDCs was then proved, by taking the bicommutative medial bimonoids and the medial bimonoid morphisms of a medial LDC.