Sometimes it is useful to think to monomorphisms as limits of a category.
Let be a monomorhism, i.e., for every pair of arrows , if , then .
This is equivalent to say that the square
is a pullback. In fact, the universal property says that, if , then there is a unique arrow such that and , so , that is, is a monomorphism. Conversely, if is a monomorphism, then the square always commutes, and it possesses the universal property since the (necessarily unique) arrow is such that and .
Dually, one characterizes epimorphisms as colimits, specifically pushouts.