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Let $X$ be a smooth manifold. I am aware of two comparison isomorphisms $H^*_{dR}(X,\mathbb{R}) \rightarrow H^*_{sing}(X,\mathbb{R})$ between de Rham cohomology and singular cohomology (with real coefficients).

The first comparison isomorphism (due to de Rham) is given by integrating smooth $k$-forms against smooth $k$-cycles (and then one has to argue that singular cohomology can be calculated with smooth $k$-chains).

The second comparison isomorphism is given by interpreting both cohomology groups as cohomology of the constant sheaf $\mathbb{R}$ on $X$. Indeed, the de Rham complex and the sheafified complex of singular cochains both give acyclic resolutions of the constant sheaf $\mathbb{R}$ on $X$.

Do these two comparison isomorphisms coincide? (A reference is plenty sufficient.)

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