A characterization of the overcoherence

Let $\mathcal{P}$ be a proper smooth formal $\mathcal{V}$-scheme, $X$ a closed subscheme of the special fiber of $\mathcal{P}$, $$\mathcal{E}$ \in F$\text{-}$D ^$\mathrm{b}$$\mathrm{c}\mathrm{o}\mathrm{h}$ (\D ^\dag{$\mathcal{P}$,$\mathbb{Q}$})$ with support in $X$. We check that $\mathcal{E}$ is $\D ^\dag {$\mathcal{P}$,$\mathbb{Q}$}$-overcoherentifandonlyif,foranymorphism$f : $\mathcal{P}$’ \to $\mathcal{P}$$ofsmoothformal$$\mathcal{V}$$-schemes,$f ^! ($\mathcal{E}$) $is$\D ^\dag{$\mathcal{P}$', $\mathbb{Q}$}$-coherent.