his analogy is apt, but some of his math is wrong.
the formula:
$h'(c) = (g \circ f)'(c) = g'(f(c))\cdot f'(c)$
is correctly expressed as:
$$\left(\lim_{f(x) \to f(c)} \dfrac{g(f(x)) - g(f(c))}{f(x) - f(c)}\right)\left(\lim_{x \to c} \dfrac{f(x) - f(c)}{x - c}\right)$$
whereas in the video he writes:
$$h'(c) = \lim_{x \to c} \dfrac{g(f(x)) - g(f(c))}{x - c} \cdot \dfrac{f(x) - f(c)}{x - c}$$
that is he has the denominator wrong in the first factor, so we can't "cancel" the f(x) - f(c) term to obtain:
$$h'(c) = \lim_{x \to c} \dfrac{g(f(x)) - g(f(c))}{x - c} = \lim_{x \to c} \dfrac{h(x) - h(c)}{x - c}$$
an important point is also glossed over:
f has to be differentiable, and differentiable means continuous, so f(x)-->f(c) as x-->c. this is KEY.
i give him credit for his vivid analogy, but algebra counts, too.