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 $X$, $Y'$ 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{Isoc} ^{\dag} (Y,X/K) \to F\text{-}\mathrm{Isoc} ^{\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{Isoc} ^{\dag\dag} (Y,X/K) $ whose objects are some particular arithmetic $\D$-modules. Furthermore, we check the equivalence of categories $\sp_{(Y,X),+} : \mathrm{Isoc} ^{\dag} (Y,X/K) \cong \mathrm{Isoc} ^{\dag\dag} (Y,X/K)$.