By definition, there exists a vertical ideal sheaf $\mathfrak{a}$ on $\mathcal{X}$ such that $\pi$ is obtained as the blow-up of $\mathcal{X}$ along $\mathfrak{a}$. The universal property of blow-ups yields a $\pi$-ample Cartier divisor $A$ on $\mathcal{X}^{\prime}$ such that $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(A)$, and $A$ is also vertical since $\mathfrak{a}$ is, which proves (i). If $\mathcal{L}$ is ample on $\mathcal{X}$ then $m\pi^{*}\mathcal{L}+A$ is ample on $\mathcal{X}^{\prime}$ for $m\gg 1$, and (ii) follows. ∎