# Initial object

<https://en.wikipedia.org/wiki/Initial_and_terminal_objects>

**An initial object** of a category 𝕮 is an object 𝑰 in 𝕮 such that for every object 𝑿 in 𝕮, there exists precisely 1 morphism from 𝑰 to 𝑿, `𝓜 : 𝑰 → 𝑿`.

*Initial objects* are also called *coterminal objects* or *universal objects*.

If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object.

A **strict initial object** `I` is one for which every morphism into `I` is an *isomorphism*.

A is the initial object:

```
     B
     ↑
D <- A -> C
     ↓
     D
```

* The empty set is the unique initial object in the category Set
* There are no zero objects in the category Set