Exercise 51.1

Proof. Let $F$ be the homotopy between $h,h^{\prime }$, and $G$ the homotopy between $k,k^{\prime }$. Let $H:X\times 1\to Z$ where $H(x,t)=G(F(x,t),t)$. Then $H(x,0)=G(F(x,0),0)=G(f(x),0)=(k\circ h)(x)$, $H(x,1)=G(F(x,1),1)=G(f(x),1)=(k^{\prime }\circ h^{\prime })(x)$.

It remains to show $H$ is continuous. $H$ is the map $(x,t)\mapsto (F(x,t),t)\mapsto G(F(x,t),t)$; since $G$ is already continuous and the composition of continuous functions is continuous, it suffices to show $(x,t)\mapsto (F(x,t),t)$ is continuous. But this is clear since this map is continuous in each coordinate in the codomain. □