Subobject classifier

In Category theory subobject classifier Ω is a special object of a category that has the following property:
for each monomorphism f: Y -> X there is a classifying morphism g: X -> Ω such that the following commutative diagram:

Y -> 1
|    |
v    v
X -> Ω

is a pullback diagram - that is, Y is the <a href="http://en2.wikipedia.org/wiki/Limit_(category_theory)">limit</a> of the diagram

        1
        |
        v
g: X -> Ω

Here 1 is the terminal object of the category.