In string diagrams,

1. A functor \(\mathscr {F} : {\cal C} \to {\cal D}\) can be represented as an edge, commonly referred to as a wire:
2. Functors compose from left to right:
3. A natural transformation \(\alpha : \mathscr {F} \to \mathscr {F}'\) can be represented as a dot on the wire from top to bottem (the opposite direction of [marsden2014category], but the same as [sterling2023models]), connecting the two functors :
4. Natural transformations (for the same pair of categories) compose vertically from top to bottem:
5. Natural transformations (from different pairs of categories) compose horizontally from left to right:
6. The two ways of composing natural transformations are related by the interchange law: i.e. \[(\alpha \mathbin {\bullet } \alpha ') \mathbin {\bullet } (\beta \mathbin {\bullet } \beta ') = (\alpha \mathbin {\bullet } \beta ) \mathbin {\bullet } (\alpha ' \mathbin {\bullet } \beta ')\]

7. The naturality in natural transformations is equivalent to the following equality: where \(X\) and \(X'\) are objects in \({\cal C}\), understood as functors from the terminal category \(\mathit {1}\) to \({\cal C}\).

   👉

   Since a string diagrams is composed from top to bottem, left to right, we can read as \[(X \mathbin {\bullet } \mathscr {F}) \mathbin {\bullet } (\alpha _X) \mathbin {\bullet } (f \mathbin {\bullet } \mathscr {G})=(X' \mathbin {\bullet } \mathscr {G}) \quad {\large =} \quad (X \mathbin {\bullet } \mathscr {F}) \mathbin {\bullet } (f \mathbin {\bullet } \mathscr {F}) \mathbin {\bullet } (\alpha '_X)=(X' \mathbin {\bullet } \mathscr {G})\] where each pair of parentheses corresponds to an overlay in the string diagram, or with the notation in the opposite direction that is more familiar to most: \[\mathscr {G}(f) \circ \alpha _X \circ \mathscr {F}(X) = \mathscr {G}(X') \quad {\large =} \quad \alpha '_X \circ \mathscr {F}(f) \circ \mathscr {F}(X) = \mathscr {G}(X')\] Note that we read the wire from \(\mathscr {F}\) to \(\mathscr {G}\) as \(\mathscr {F}\) before the natural transformation, but as \(\mathscr {G}\) after the transformation.

   Effectively naturality says that the natural transformation and function \(f\) “slide past each other”, and so we can draw them as two parallel wires to illustrate this.