# Category Theory/Subcategories

**Definition (subcategory)**:

Let be a category. Then a **subcategory** of is a category such that and .

**Definition (full)**:

A subcategory of a category is called **full** iff for all , we have

- .

**Proposition (limits are preserved when restricting to a full subcategory)**:

Let be a category, let be a diagram in , and let be a full subcategory of . Suppose that is a limit over in such that and all targets of the are in . Then is a limit over in .

**Proof:** Certainly, the underlying cone of is contained within , because the subcategory is full. Now let another cone in over the diagram (which, analogously, is a diagram in ) be given. By the universal property of in , there exists a unique morphism which satisfies for all . Since is full, is in .

Analogously, we have:

**Proposition (colimits are preserved when restricting to a full subcategory)**:

Let be a category, let be a diagram in , and let be a full subcategory of . Suppose that is a colimit over in such that and all domains of the are in . Then is a colimit over in .

**Proof:** This follows from its "dual" proposition, reversing all arrows in its statement and proof except the direction of the diagram functor.