  The study of abstraction and composition - the focus of category theory -
naturally leads to sophisticated diagrams which can encode complex algebraic
semantics. Consequently, these diagrams facilitate a clearer visual
comprehension of diverse theoretical and applied systems. Complex algebraic
structures - otherwise represented by a forest of symbols - can be encoded into
diagrams with intuitive graphical rules. The prevailing paradigm for
diagrammatic category theory are monoidal string diagrams whose specification
reflects the axioms of monoidal categories. However, such diagrams struggle in
accurately portraying crucial categorical constructs such as functors or
natural transformations, obscuring central concepts such as the Yoneda lemma or
the simultaneous consideration of hom-functors and products. In this work, we
introduce functor string diagrams, a systematic approach for the development of
categorical diagrams which allows functors, natural transformations, and
products to be clearly represented. We validate their practicality in multiple
dimensions. We show their usefulness for theoretical manipulations by proving
the Yoneda lemma, show that they encompass monoidal string diagrams and hence
their helpful properties, and end by showing their exceptional applied utility
by leveraging them to underpin neural circuit diagrams, a method which, at
last, allows deep learning architectures to be comprehensively and rigorously
expressed.
