The definition couldn’t be simpler. We really only need to define the image of a singular -cube in and extend by linearity. And since is a function, we can just compose it with to get a singular -cube . What’s the face of this singular -cube? Why it’s
and so we find that this map commutes with the boundary operation , making it a chain map.
We should still check functoriality. The identity map clearly gives us the identity chain map. And if and are two smooth maps, then we can check
which makes this construction a covariant functor.