**Definition 1**. Let be a direct system in a category Suppose there exists an object and morphisms for every such that the following conditions are satisfied:

1) whenever

2) (The universal property) Suppose that there exist an object and morphisms such that whenever Then there exists a unique morphism such that

Then is called the **direct limit** of our direct system and we will just write

**Definition 2**. Let be an inverse system in a category Suppose that there exists an object and morphisms for every such that the following conditions are satisfied:

1) whenever

2) If there exists some object and morphisms for every such that whenever then there must exist a unique morphism such that for all

Then is called the **inverse limit** of our inverse system and we will just write

In our definitions I wrote “the” direct limit and “the” inverse limit. That is because “the” direct (resp. inverse) limit of a direct (resp. inverse) system, if it exists, is unique, up to isomorphism of course, as we are going to see:

**Fact**. The direct (resp. inverse) limit of a direct (resp. inverse) system is unique up to isomorphism if it exists.

*Proof.* I will prove the fact for direct limit only. The proof for inverse limit is similar. Suppose that both and satisfy the conditions in Dedinition 1. Then, by condition 2), there exist (unique) morphisms and such that and Thus the morphism satisfies But we also have that the identity morphism satisfies Thus, by the uniqueness condition in 2), we must have Similarly and hence is an isomorphism and is its inverse.

**Remark.** You might have found the relationships between the morphisms in the definition of direct (resp. inverse) systems and limits complicated but they are not! If you draw diagrams, then you will see that they just mean that those diagrams are commutative.