Exercise 6.2.20

Theorem: Fix a small category . Let be a locally small category with pullbacks. For functors , a natural transformation is monic in if and only if is monic for all .

Proof: Suppose is monic for all . Then, suppose we have so that . Then for all . But then for all and . So is monic.

Now suppose that is monic as a map in . Then, the following square is a pullback:

The evaluation functor preserves limits, so we have that:

is a pullback for each . Suppose we have maps with . Then is a cone over . But then the pullback square above implies that , and is monic.