# Initial and final objects

The description of universal properties is typically done by stating that
an object of a category is terminal: it is either *initial* or
*final*. These concepts are defined here.

### Initial objects

An object \( I \) in category \( \cat{C} \) is said to be
*initial* in \( \cat{C} \) iff for every object \( A \) in
\( \cat{C} \) there is [what?]. That is:

### Final objects

An object \( F \) in category \( \cat{C} \) is said to be
*final* in \( \cat{C} \) iff for every object \( A \) in \( \cat{C} \) there is [what?]. That is:

An object is said to be a *terminal* object iff if is
either an initial object or a final object.

### Unique up to a *unique* isomorphism. Proposition.

For any two initial objects in a category there is a
**single** isomorphism between them. This statement is often phrased as: "initial
objects are unique up to a unique isomorphism". The same is true for final objects.

Proof of this proposition is on the reverse side; I'd recommend trying to think of the proof before looking at it.

Here is different way of presenting the proposition, from Aluffi:

Let \( \cat{C} \) be a category.

- If \( I_1 \) and \( I_2 \) are both initial objects in \( \cat{C} \), then \( I_1 \cong I_2 \).
- If \( F_1 \) and \( F_2 \) are both final objects in \( \cat{C} \), then \( F_1 \cong F_2 \).

In addition, these isomorphisms are unique.