Uniqueness of the blowup

By [F2] the blowup $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X\to X$ is an object of the category of $X$-schemes in which the inverse image of $Z$ is an effective Cartier divisor, and by hypothesis $Y$ is such an object as well.

Applying the universal property of the blowup [F1] to the $X$-scheme $Y$ gives a unique $X$-morphism $u\text{\hspace{0.17em}⁣}:Y\to {\mathrm{Bl}}_{\mathcal{I}}X$ with $\pi \circ u={\pi }^{\mathrm{\prime }}$; applying the universal property carried by $Y$ to the $X$-scheme ${\mathrm{Bl}}_{\mathcal{I}}X$ gives a unique $X$-morphism $v\text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X\to Y$ with ${\pi }^{\mathrm{\prime }}\circ v=\pi$.

The composite $v\circ u\text{\hspace{0.17em}⁣}:Y\to Y$ is an $X$-morphism with ${\pi }^{\mathrm{\prime }}\circ (v\circ u)={\pi }^{\mathrm{\prime }}$, and so is ${\mathrm{id}}_Y$; since by hypothesis there is at most one $X$-morphism from the admissible $X$-scheme $Y$ to $Y$, namely the map required by the universal property, we get $v\circ u={\mathrm{id}}_Y$; symmetrically $u\circ v={\mathrm{id}}_{{\mathrm{Bl}}_{\mathcal{I}}X}$ because $u\circ v$ and the identity are both $X$-morphisms from ${\mathrm{Bl}}_{\mathcal{I}}X$ to itself and [F1] gives a unique one. Hence $u$ is an $X$-isomorphism, and it is the unique one: any $X$-isomorphism $Y\to {\mathrm{Bl}}_{\mathcal{I}}X$ is an $X$-morphism between admissible objects and therefore equals $u$ by the uniqueness clause of [F1]; in particular any two models of the blowup are uniquely isomorphic over $X$.