**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.