Does there exist $y\in[0, 1]$, such that $f(y) = g(y)$ for each pair of functions $f$ and $g$ under specific conditions?

Question

Suppose $f, g: [0,1] \rightarrow [0,1]$ are continuous functions, such that for each $x$ from $[0,1]$ the identity $f(g(x)) = g(f(x))$ is true. Does there always exist $y\in[0, 1]$, such that $f(y) = g(y)$?

That statement seems to be true, but I do not know how to prove it.

Any help will be appreciated.