After the effort invested in proving the general theorem on acyclic models, it is time to apply it to topology. First, let us prove:

Theorem 5Suppose are homotopic. Then the maps are equal.

*Proof:* Suppose is a homotopy with . Then the maps factor as

so if we show that the inclusions sending to induce equal maps on homology, we will be done.

Write for simplicity. For each space , the maps are *natural*. More precisely, if are the two functors , then are natural transformations between them.