Reuben - very good observation! There's a general concept of [adjunction](https://ncatlab.org/nlab/show/adjunction#idea) that includes adjoint functors _between_ categories and caps and cups _within_ monoidal categories.
An adjunction can live in any [2-category](https://ncatlab.org/nlab/show/2-category). A 2-category is a thing with
* morphisms, and
Adjoint functors live in the 2-category of
* functors, and
* natural transformations
But a monoidal category is the same as a 2-category with one object, so adjunctions can live in there too!
Click to see the definition of 'adjunction', then scroll down to see the definition in terms of string diagrams, and you'll see the snake equations (also known as zig-zag equations).
You really need to learn about 2-categories to penetrate the deeper layers of category theory: the thing of all sets is best thought of as a category. While there's a category of (small) categories, the thing of all categories is best thought of as a 2-category.
Luckily, 2-categories lend themselves to nice 2-dimensional pictures. I think the most fun gentle introduction is these videos:
* The Catsters, [string diagrams](https://www.youtube.com/watch?v=USYRDDZ9yEc&list=PL50ABC4792BD0A086).