Cardinality of a collection of functions whose composition commutes

Question

Consider a collection of functions $\mathcal F$ where (i) each individual element is a strictly increasing function from [0,1] to [0,1]; (ii) for any $f,g$ $f \le g$ or $f \ge g$ and $f\ne g$; and (iii) for any finite subcollection of functions $f_{1}\circ f_{2}\circ f_{3}\circ....\circ f_{n}=f_{\pi(1)}\circ f_{\pi(2)}\circ f_{\pi(3)}\circ....\circ f_{\pi(n)}$, where $\pi$ is permutation of the first $n$ natural numbers, that is any finite collection of functions satisfies the commutative property under composition. Can such a class of functions be uncountable?