A Fermat theory is an extension of a Lawvere theory which adds Hadamard’s Lemma as an axiom. In turn, this allows us to differentiate maps in a Fermat theory, and thus every Fermat theory is a Cartesian differential category. In this talk, we introduce Cartesian Fermat categories, which are briefly Fermat theories that are not necessarily Lawvere theories (and do not assume multiplication). Still, every Cartesian Fermat theory is a Cartesian differential category, where moreover, the linear maps correspond precisely to the additive maps. We will also give a friendly introduction to Cartesian differential categories, and discuss future ideas for Cartesian Fermat categories.