Bending, Yanking, and Cartesian Squares in Double Categories

This article delves into advanced category theory concepts using string diagrams, specifically within double categories and proarrow equipments. It explains the yanking identity for companions and conjoints, demonstrating how zig-zag arrows can be simplified. The spider lemma is introduced as a powerful tool for bending arrows in diagrams, with a proof using units and counits. Finally, the concept of cartesian squares in double categories is discussed as a universal construction, analogous to categorical products. The content is highly technical, focusing on mathematical structures relevant to computer science and programming language theory.