<p>This paper establishes a foundational connection between optimization and category theory, offering a novel framework that bridges these two disciplines. In the first step, we use posets to reveal the extremal nature underlying fundamental categorical concepts, demonstrating how extremality serves as a unifying concept in category theory. We then extend this perspective by introducing a preorder relation between functors, providing a fresh interpretation of Mac Lane’s celebrated statement, “all concepts are Kan extensions,” by rephrasing it as “all concepts are extremals.” We show, through examples, that optimization naturally aligns with categorical structures, notably by proving that the convex envelope operator defines a functor arising as a left Kan extension.</p>