The Stacks project

Proof. Let $(\mathcal{F}_ n)$ and $(\mathcal{G}_ n)$ be objects of $\textit{Coh}(X, \mathcal{I})$. A morphism of pro-objects $\alpha $ from $(\mathcal{F}_ n)$ to $(\mathcal{G}_ n)$ is given by a system of maps $\alpha _ n : \mathcal{F}_{m(n)} \to \mathcal{G}_ n$ compatible with the transition maps where $\mathbf{N} \to \mathbf{N}$, $n \mapsto m(n)$ is an increasing function (in particular $m(n) \geq n$). Since $\mathcal{F}_ n = \mathcal{F}_{m(n)}/\mathcal{I}^ n\mathcal{F}_{m(n)}$ and since $\mathcal{G}_ n$ is annihilated by $\mathcal{I}^ n$ we see that $\alpha _ n$ induces a map $\mathcal{F}_ n \to \mathcal{G}_ n$. $\square$