Monomorphisms as limits

Sometimes it is useful to think to monomorphisms as limits of a category.

Let $f\colon A \to B$ be a monomorhism, i.e., for every pair of arrows $g,h\colon C \to A$ , if $f \circ g = f \circ h$ , then $g = h$ .

This is equivalent to say that the square

is a pullback. In fact, the universal property says that, if $f \circ g = f \circ h$ , then there is a unique arrow $!\colon C \to A$ such that $g = 1_A \circ !$ and $h = 1_a \circ !$ , so $! = g = h$ , that is, $f$ is a monomorphism. Conversely, if $f$ is a monomorphism, then the square always commutes, and it possesses the universal property since the (necessarily unique) arrow $g = h$ is such that $1_A \circ g = g$ and $1_A \circ h = h$ .