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
Final objects
An object
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
- If
and are both initial objects in , then . - If
and are both final objects in , then .
In addition, these isomorphisms are unique.