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I'll just add that one immediate neat implication of making the general abstract comonad theory behind this explicit is that it gives in full generality that for any topos (or $\infty$-topos) $\mathbf{H}$ equipped with an "infinitesimal shape modality" $X\mapsto \Im X = X_{dR}$, then since $\mathrm{Jet} := (i_X)^\ast (i_X)_\ast \colon \mathbf{H}_{/X} \to \mathbf{H}_{/X}$ is a right adjoint, a standard fact (here for toposes, here maybe for $\infty$-toposes) gives that its EM-category, hence the category $\mathrm{PDE}(X)$ of PDEs in $X$ is itself a topos, sitting by a geometric morphism

I'll just add that one immediate neat implication of making the general abstract comonad theory behind this explicit is that it gives in full generality that for any topos (or $\infty$-topos) $\mathbf{H}$ equipped with an "infinitesimal shape modality" $X\mapsto \Im X = X_{dR}$, then since $\mathrm{Jet} := (i_X)^\ast (i_X)_\ast \colon \mathbf{H}_{/X} \to \mathbf{H}_{/X}$ is a right adjoint, a standard fact (here for toposes, here maybe for $\infty$-toposes) gives that its EM-category, hence the category $\mathrm{PDE}(X)$ of PDEs in $X$ is itself a topos, sitting by a geometric morphism

over $\mathbf{H}$, whose inverse image is (non-fully) that co-Kleisli category of bundles over $X$ with differential operators between them.

over $\mathbf{H}$, whose direct image is (non-fully) that co-Kleisli category of bundles over $X$ with differential operators between them.

Of course the Kleisli category in question is the full subcategory of that, and the statement that I asked for is the special case of the above made fully explicitly in

Of course the Kleisli category in question is a full subcategory of that, and the statement that I asked for is the special case of the above made fully explicitly in