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.