notation. string diagrams: adjunction [nakahira2023diagrammatic, sec. 3.1] [tt-001U]

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Here we follow the string diagram style of [marsden2014category] and [sterling2023models], but with additional string diagram types inspired by [nakahira2023diagrammatic, eq. 3.1, 4.3].

1. The covariant hom-functor \({\cal C}(X, -)\), denoted \(-^X\), can be represented in string diagrams as where the dotted circle denotes any arrows with the domain \(X\) and codomain \(-\).
2. The contravariant hom-functor \({\cal C}(-, X)\), denoted \(X^-\), can be represented in a similar manner.
3. The hom-bifunctor \({\cal C}(-,=)\), also denoted \(=^-\), can be represented as where the dotted circle denotes any arrows with the domain \(-\) and codomain \(=\).
4. The natural isomorphism \[{\cal D}(\mathscr {L}(-), =) \cong {\cal C}(-, \mathscr {R}(=))\] in adjunction can be represented as