**Exercise 6.1.21**

**Theorem**:
For a locally small category and , if are presheaves with , then either or is the trivial presheaf .

*Proof*:
A natural isomorphism amounts to a universal element .
Take an arbitrary and .
Universality says we have a unique map with .

Now either or .
Suppose that it is in .
Then for *any* map we have:

from the behavior of sums of maps in .
But, this means that is always in .
Thus, if we picked a , there would be *no* map whose image under carried to , violating universality – thus, there must be no elements in .
Since was arbitrary, we have for all – i.e., is the constant presheaf .

Supposing and using a symmetric argument shows that . Thus, one of or must be the trivial presheaf.