# 3.4. Adjunctions and data migration#

## 3.4.1. Pulling back data along a functor#

### Exercise 3.67.#

Every arrow indicates âthis 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:

Reveal 7S answer

## 3.4.2. Adjunctions#

### 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). Youâll 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 authorâs definition (and the objects Y correspond to the objects c). In the diagram in Wikipediaâs defintion, youâll 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 âisomorphismâ rather than a morphism. We call \(f\) and \(g\) âinversesâ but more specifically they are inverse morphisms (or âobjectâ isomorphisms) (not inverse functors, which act between categories).

See instead Remark 3.59 and the concept of an Equivalence of categories for examples of âinverseâ 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 âtrueâ inverse functor.

An adjunction relaxes the concept of âinverseâ 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 âmiddle groundâ that is used to define a natural relationship.

### Example 3.71.#

### Example 3.72.#

See also Exponential object.

### Exercise 3.73.#

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

Reveal 7S answer

### Example 3.74.#

## 3.4.3 Left and right pushforward functors, ÎŁ and Î #

## 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.

Reveal 7S answer

### Exercise 3.78.#

Reveal 7S answer