**Exercise 5.1.41**

**Theorem**:
A map is monic if and only if the square:

is a pullback.

*Proof*:
Assume the square is a pullback.
Let be maps with .
Then is a cone over .
Thus, we have a unique with and , so .

Assume is monic. Let be an arbitrary cone over . We need a unique such that:

commutes. Note that since is a cone, we have and since is monic, . Thus, taking makes the diagram commute. Clearly then, this is the unique such map, because if any other made the diagram commute, we would have .