Blow-up of morphisms

Suppose we have a morphism . Let’s say we are going to blow up along a coherent sheaf of ideals, . We can take the inverse image ideal sheaf of in . Blow up along : . Whence lifts to a morphism in the obvious sense.
Furthermore, as a result, the ideal sheaves defining the exceptional divisors are both inverse image ideal sheaves of the , hence they correspond as well according to the lift , i.e, pull back the exceptional divisor on to the exceptional divisor on .

Such a lift exists uniquely, according to the universal property of the blowing up projection : the inverse image ideal sheaf is invertible.