Illustration of homotopies

Question

While there are many pictures path homotopies, I fail to find any that illustrate normal homotopies (in the event "a normal homotopy" is something else, I clarify that I mean given two continuous functions $f$, $g\colon X \to Y$ a function $H(x,t)\colon X \times I \to Y$ such that $H(x,0) = f(x)$ and $H(x,1) = g(x)$.) While the intuitive notion of "continuous deformation as $t$ increases" is clear, a picture would probably help my intuition further.

Answer 1

This answer is kind of going off of what Qiaochu said in the comments. It's beneficial for me to think of this homotopy in terms of the mapping cylinders $Z_f$ and $Z_g$:

If you go with this viewpoint, then the maps $f, g:X\to Y$ become the inclusions of $X$ into the two cylinders $Z_f$ and $Z_g$ via $X\hookrightarrow X\times\{1\}$. A homotopy between $f$ and $g$ would correspond in the picture to a continuous deformation of how the base of the cylinder $X\times I$ is "glued" onto $Y$. I hope this helps!