A pasting diagram represents categories as points (0-dimensional), arrows \(\to \) (1-dimensional), natural transformations as surfaces with level-2 arrows \(\Rightarrow \) (0-dimensional).

For example:

It's dual to a corresponding string diagram

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.

The functor category from \({\cal C}\) to \({\cal D}\), denoted \([{\cal C}, {\cal D}]\) or \({\cal D}^{{\cal C}}\), is a category whose objects are functors from \({\cal C}\) to \({\cal D}\) and whose arrows are natural transformations between them, where composition is given by and the identity is given by

One can think of a functor category \([{\cal C}, {\cal D}]\) or \({\cal D}^{{\cal C}}\) as a category of diagrams in \(D\) indexed (or labelled) by the objects from \({\cal C}\).

This particularly makes sense in a diagram.

A natural transformation \(\sigma : \mathscr {F} \to \mathscr {G}\) between functors \(\mathscr {F}: {\cal C} \to {\cal D}\) and \(\mathscr {G} : {\cal C} \to {\cal D}\) is called a natural isomorphism or a natural equivalence, denoted \(\sigma : \mathscr {F} \cong \mathscr {G}\), if each component \(\sigma _X : \mathscr {F}(X) \to \mathscr {G}(X)\) is an isomorphism in \({\cal D}\), i.e. \(\mathscr {F}(X) \underset {\sigma _X}{\cong } \mathscr {G}(X)\).

We call \(\mathscr {F}\) and \(\mathscr {G}\) naturally isomorphic to each other.

We also say that \(\mathscr {F}(X) \cong \mathscr {G}(X)\) naturally in \(X\) [leinster2016basic, 1.3.12].

Diagramatically,

A natural isomorphism between functors from categories \({\cal C}\) and \({\cal D}\) is an isomorphism in the functor category \([{\cal C}, {\cal D}]\).

The cateories \({\cal C}\) and \({\cal D}\) are called isomorphic, denoted \({\cal C} \cong {\cal D}\), iff there exists functors such that \[\mathit {1}_{{\cal C}}=\mathscr {F} \mathbin {\bullet } \mathscr {G}\] and \[\mathit {1}_{{\cal D}}=\mathscr {G} \mathbin {\bullet } \mathscr {F} \]

The categories \({\cal C}\) and \({\cal D}\) are called equivalent, denoted \({\cal C} \simeq {\cal D}\), iff there exist functors together with natural isomorphisms \[\mathit {1}_{{\cal C}} \cong \mathscr {F} \mathbin {\bullet } \mathscr {G}\] and \[\mathscr {G} \mathbin {\bullet } \mathscr {F} \cong \mathit {1}_{{\cal D}} \]