The isomorphism to Čech cohomology as evaluation on the Čech nerve

We describe explicitely the unique isomorphism ${\mathrm{H}}_{sin}^{\ast }(X,\underset{‾}{\mathbb{Z}})\stackrel{\sim }{\to }{\check{\mathrm{H}}}_{\mathcal{U}}^{\ast }(X,\underset{‾}{\mathbb{Z}})$ between the cohomologies computed with the singular and Čech acyclic sheaf resolutions as the evaluation, up to sign, of singular cohomology classes at the simplices of the Čech nerve $N\mathcal{U}$ through a homotopy equivalence $N\mathcal{U}\to X$.