Full faithfulness without Frobenius structure and partially overcoherent isocrystals

Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field $k$. Let $X$ be a variety over $k$, $Y$ be an open of $X$, $Y^{\mathrm{\prime }}$ be an open of $Y$ dense in $X$. We extend Kedlaya’s full faithfulness as follows (we do not suppose $Y$ to be smooth): the canonical functor $F$\text{-}$$\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{c}$ ^{\dag} (Y,X/K) \to F$\text{-}$$\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{c}$ ^{\dag} (Y,Y/K) $ is fully faithfull. Suppose now $Y$ smooth. We construct the category of partially overcoherent isocrystals over $(Y,X)$ denoted by ${\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{c}}^{\mathrm{†}\mathrm{†}}(Y,X\mathrm{/}K)$ whose objects are some particular arithmetic $\text{D}$-modules. Furthermore, we check the equivalence of categories $\sp_{(Y,X),+} : $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{c}$ ^{\dag} (Y,X/K) \cong $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{c}$ ^{\dag\dag} (Y,X/K)$.