**APPENDIX II**

**Adjoint functors with examples from graphs and dynamical systems**

Much mathematical struggle (e.g. ‘solving equations’) aims to partially invert a given transformation. In particular, in the case of a functor Φ from one category to another, Kan (1958) noticed that there is sometimes a uniquely determined functor in the opposite direction that, while not actually inverting Φ, is the ‘best approximate inverse’ in either a left-or a right-handed sense. The given functor is typically so obvious that one might not have mentioned it, whereas its resulting adjoint functor is a construction bristling with content that moves mathematics forward. ...

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