Definition

We can now think of functors between two fixed categories as objects worthy of study, with natural transformations as the arrows between them.

One can verify that $D^C$ does indeed satisfy the axioms of a category.

Examples

The main examples most people stumble across are probably Universal Properties III - Yoneda's Lemma and presheaves (in algebraic geometry), but for now we will satisfy ourselves with some much simpler examples.

Functors from the categories $\mathbf{\text{0}}$ and $\mathbf{\text{1}}$

Recall that the empty category $\mathbf{\text{0}}$ has no objects and no arrows. For each category $C$ there is a unique functor $\mathbf{\text{0}}\to C$, namely the empty functor (with empty object map and empty arrow map). It is straightforward to verify we have an equivalence of categories $C^{\mathbf{\text{0}}}\simeq \mathbf{\text{1}}.$

Similarly, the category $\mathbf{\text{1}}$ has a single object and only the identity arrow on that object. Convince yourself that functors $\mathbf{\text{1}}\to C$ are in bijection with objects of $C$. Moreover, the natural transformations between such functors are in bijection with arrows in $C$. You should now be able to verify we have an equivalence of categories $C^{\mathbf{\text{1}}}\simeq C$.

Commutative diagrams of a fixed shape

Suppose $J$ is a fixed category. Recall that for any category $C$, functors $F:J\to C$ can be thought of as "commutative diagrams in $C$ of shape $J$." For example, suppose $J$ is the category with three objects and two nonidentity arrows, as illustrated below:

Then each functor $F:J\to C$ corresponds to a diagram in $C$ of the form

This functor category is useful when studying pullbacks and pushforwards.

Suggested next note

Universal Properties III - Yoneda's Lemma