Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let C be the full subcategory of (Sch/X) consisting of Y → X such that the inverse image of Z is an effective Cartier divisor on Y. Then the blowing up b : X′→X of Z in X is a final object of C. See Lemma Tag 0806.

Slogan: Universal property of blowing up