Find me a sigmoid function with fixed point at the point of inflection in the unit interval

Question

I am interested in finding sigmoid (S-shaped) functions $f$ on the unit interval $[0,1]$ such that

1. For some $a \in (0, 1)$, $f(a) = a$ and $f'(a^-) > 0$ and $f'(a^+) < 0$.

2. $f(0) = 0$ and $f(1) = 1$

The last criteria is trivial to achieve by rescaling in the y-axis, but I am curious on whether there are any functions that satisfy the first criteria and also naturally satisfy the last criteria.