Homeomorhism of pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})$

Question

I'm having trouble following an argument in Bredon's Topology and Geometry. He says that the pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})$ are homeomorphic.

If I am recalling the definition correctly: we say that two pairs $(X,A)$ and $(Y,B)$ are homeomorphic if there exists an homeomorphism $f \colon X \to Y$ which restricts to a homeomorphism $F_{|A} \colon A \to B$.

He arguments using the following picture:

I think my problem is that I don't follow the step from the second picture to the third one. If I am not misunderstanding the second diagram (which I'm afraid I am), then I don't see how he obtains the third picture.

Any help would be appreciated. Thanks in advance!