Localization is left adjoint to global sections

On an affine chart $U=\mathrm{Spec}R$, put $Q=\mathrm{\Gamma }(U,{\mathcal{D}}_{\kappa })$. Right multiplication by $A_{\kappa }$ commutes with left multiplication by $R$. Thus $Q{\otimes }_{A_{\kappa }}V$ is an $R$-module, and for every $f\in R$ there is a natural isomorphism $Q_f{\otimes }_{A_{\kappa }}V\cong (Q{\otimes }_{A_{\kappa }}V{)}_f,(q\mathrm{/}{f}^m)\otimes v⟼(q\otimes v)\mathrm{/}{f}^m\mathrm{.}$ The inverse sends $(q\otimes v)\mathrm{/}{f}^m$ to $(q\mathrm{/}{f}^m)\otimes v$; additivity and the balancing relations make both maps well-defined. By [F2] and [F3], the tensor presheaf on the distinguished-open basis of $U$ is therefore the associated sheaf of $Q{\otimes }_{A_{\kappa }}V$. Its sheafification has that restriction on $U$, proving that ${\mathrm{Loc}}_{\kappa }V$ is quasi-coherent. Left multiplication on its first factor gives its left ${\mathcal{D}}_{\kappa }$-action.

The tensor-Hom correspondence here follows directly from balancing. Given an $A_{\kappa }$-linear map $V\to {\mathrm{Hom}}_{{\mathcal{D}}_{\kappa }}({\mathcal{D}}_{\kappa },M)$, write its value at $v$ as $h_v$ and send $d\otimes v$ over $W$ to $h_v{\mathrm{\mid }}_W(d)$. The $A_{\kappa }$-action on Hom is $(ah)(d)=h(da)$, so the rule is balanced, respects restrictions, and extends through sheafification to a ${\mathcal{D}}_{\kappa }$-linear map. Conversely such a sheaf map gives $h_v$ by $d\mapsto \varphi (d\otimes v)$. These constructions are inverse and natural, and step 1.1 places their source in the required quasi-coherent category.

Evaluation at $1$ identifies ${\mathrm{Hom}}_{{\mathcal{D}}_{\kappa }}({\mathcal{D}}_{\kappa },M)$ with $\mathrm{\Gamma }(X,M)$. A morphism is determined by the image of $1$; conversely, each global section $m$ defines the morphism $d\mapsto dm$. Under this identification the $A_{\kappa }$-action is the one from [F1]. Substituting into step 2.1 gives the natural bijection ${\mathrm{Hom}}_{{\mathcal{D}}_{\kappa }}({\mathrm{Loc}}_{\kappa }V,M)\cong {\mathrm{Hom}}_{A_{\kappa }}(V,{\mathrm{\Gamma }}_{\kappa }M),$ which is the asserted adjunction. Right exactness of localization follows because a left adjoint preserves cokernels.