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 .