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

Definition of a functor category

Suppose $C$ and $D$ are two categories. We denote the category of functors from $C$ to $D$ by ${D}^{C}$. The objects of this category are functors $F:C\beta \x86\x92D$, and the arrows of this category are natural transformations $\mathrm{{\rm O}\x84}:F\beta \x87\x92G$ between such functors.

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

Why this notation?

You might be wondering why we use this exponential-style notation for functor categories. The first few examples below should hint at a reason. They can eventually be made more formal with the categorical concept of an "exponential object."

Examples

The main examples most people stumble across are probably 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 $\mathbf{\text{0}}\beta \x86\x92C$, namely the empty functor (with empty object map and empty arrow map). It is straightforward to verify we have an equivalence of categories ${C}^{\mathbf{\text{0}}}\beta \x89\x83\mathbf{\text{1}}.$

Similarly, the category $\mathbf{\text{1}}$ has a single object and only the identity arrow on that object. Convince yourself that functors $\mathbf{\text{1}}\beta \x86\x92C$ 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 ${C}^{\mathbf{\text{1}}}\beta \x89\x83C$.

Commutative diagrams of a fixed shape

Suppose $J$ is a fixed category. Recall that for any category $C$, functors $F:J\beta \x86\x92C$ 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: