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.