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Let $h:X\to Y$ be a separated morphism of adic spaces of finite type over a non-archimedean field $k$ with $Y$ affinoid and of dimension $\le 1$, let $L$ be a locally closed constructible subset of $X$ and let $g:(X,L)\to Y$ be the morphism of pseudo-adic spaces induced by $h$. Let $A$ be a noetherian torsion ring with torsion prime to the characteristic of the residue field of the valuation ring of $k$ and let $F$ be a constant $A$-module of finite type on ${(X,L)}_{\acute{\mathrm{e}}\mathrm{t}}$. There is a natural class $𝒞\left(Y\right)$ of $A$-modules on $Y_{\acute{\mathrm{e}}\mathrm{t}}$ generated by the constructible $A$-modules...