I came across a function while trying to show that for two pairs of homotopic maps, $h,h'$ and $k,k'$, the meaningful compositions, $k\circ h, k'\circ h'$, are also homotopic. But, my question is just regarding continuity and should be independent from this topic.
Let $I=[0,1]$ and $X$, $Y$ and $Z$ be topological spaces. Define the continuous functions: $F:X\times I\to Y$ and $G:Y\times I\to Z$. Show that $H(x,t):=G(F(x,t),t): X\times I\to Z$ is a continuous function.
Here's my attempt: Naturally we want to show that for $U_Z$ open in $Z$ we have $H^{-1}(U_Z)$ open in $X\times I$. I show that for $(x,t)\in H^{-1}(U_Z)\ \exists\ B_X,\ B_I$ basis elements in $X$, $I$ respectively s.t. $(x,t)\in B_X\times B_I\subset H^{-1}(U_Z)$. Since $G$ is continuous, $G^{-1}(U_Z)$ is open in $Y\times I$, therefore $\exists\ B_Y, B_{I,1}$ basis elements in $Y$, $I$ respectively where $(F(x,t),t)\in B_Y\times B_{I,1}\subset G^{-1}(U_Z)$. By a similar logic $\exists\ B_X, B_{I,2}$ basis elements in $X$, $I$ respectively s.t. $(x,t)\in B_X\times B_{I,2}\subset F^{-1}(B_Y)$. By property of a basis, $\exists\ B_I$ s.t. $(x,t)\in B_I\subset B_{I,1}\cap B_{I,2}$. So I've shown that $H^{-1}(U_Z)$ is open in $X\times I$ with $B_X\times B_I$.
I tried to find the same question on this website and only found this post (https://math.stackexchange.com/a/1832186/1100158) saying that continuity is a trivial matter. Is there a simpler way than my approach? I struggled proving this, so I'm not sure I did it in the best/simpler/shorter way.