Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = \int_{\partial M} \omega.$$ Doesn't that look like a naturality condition in the sense of category theory? Somehow, integration is natural with respect to boundaries (or vice versa?). Can we make this precise?

What I have tried so far: If $\Omega_0^k(M)$ denotes the vector space of compactly-supported differential forms of degree $k$ on $M$, and $d : \partial M \hookrightarrow M$ denotes the inclusion of the boundary, Stokes' Theorem says that the diagram $$ \require{AMScd} \begin{CD} \Omega_0^{n-1}(M) @>{d}>> \Omega_0^n(M) \\ @Vd^*VV @VV{\int_{M}}V \\\ \Omega_0^{n-1}(\partial M) @>{\int_{\partial M}}>> \mathbb{R} \end{CD} $$

commutes. Is that correct? (I'm not sure about the $d^*$). This looks more like dinaturality, but I am not sure how to make a precise connection. Perhaps the cobordism category will be useful?

Any other categorical interpretation of Stokes' Theorem would also be appreciated. Notice that such interpretations are by no means useless, a priori, and could perhaps even lead to more conceptual proofs. See for instance

Roeder, David. "Category theory applied to Pontryagin duality." Pacific Journal of Mathematics 52.2 (1974): 519-527.

Hartig, Donald G. "The Riesz representation theorem revisited." American Mathematical Monthly (1983): 277-280.