Can someone give me a geometrical interpretation (picture) of mapping cylinder of a continuous map $g: X \to Y$, where $\operatorname{Cyl}(g) = Z \cup_f Y$, where $Z = X \times [0, 1],\, A = X \times \{0\}$ and $f$ is $g$ composed with the obvious identification $X \times \{0\} \cong X$.
Or its mapping cone, where $\operatorname{Cone}(g) = \operatorname{Cyl}(g)/{\sim}$, where $\sim$ is the smallest equivalence relation such that $(x, 1) \sim (x', 1)$ for all $x, x' \in X$.
A graduation cap is an example of a mapping cylinder $g : X \to Y$ where $X = S^1$, $Y = [-2,2] \times [-2,2]$, and $g$ is the inclusion map.