# 3.4. Adjunctions and data migration#

## 3.4.1. Pulling back data along a functor#

### Exercise 3.67.#

Every arrow indicates âthis is the state that I came fromâ (the source). You could use this diagram to work back to how you could have gotten to some final state:

### Definition 3.70.#

Read through Adjoint functors and in particular the section Definition via Hom-set adjunction for a more detailed version of this definition. Unfortunately, the Wikipedia article describes the left adjoint from đ â đ (opposite the authors). Youâll have to switch these two letters everywhere to read it. In particular, this switch implies the objects X in the Wikipedia definition correspond to the objects d in the authorâs definition (and the objects Y correspond to the objects c). In the diagram in Wikipediaâs defintion, youâll need to switch the morphisms f and g, as well as reflect the whole diagram across a vertical line in the center of the diagram.

If two morphisms are inverses, then $$fâ¨žg = id_A$$ and $$gâ¨žf = id_B$$, and we say that A and B are isomorphic objects. We call $$f$$ more specifically an âisomorphismâ rather than a morphism. We call $$f$$ and $$g$$ âinversesâ but more specifically they are inverse morphisms (or âobjectâ isomorphisms) (not inverse functors, which act between categories).

See instead Remark 3.59 and the concept of an Equivalence of categories for examples of âinverseâ functors. Strictly speaking, an equivalence of categories only requires that isomorphic objects in the first category correspond to isomorphic objects in the second category (think of a map between preorders rather than partial orders). An Isomorphism of categories would require a âtrueâ inverse functor.

An adjunction relaxes the concept of âinverseâ even more than an equivalence of categories does. It complicates the concept of a Hom-functor (a single bifunctor) by effectively introducing two bifunctors (that now act on different categories) to the category of sets. The category of sets provides a âmiddle groundâ that is used to define a natural relationship.

### Exercise 3.73.#

For 3. we are defining a new function that adds 3. That is, p(3) := (b) âŚ 3 + b

## 3.4.4. Single set summaries of databases#

### Exercise 3.76.#

This functor sends all objects to the object 1 and all morphisms to idâ.