# Monomorphisms as limits

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.

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