Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's).

Notice that given a fixed number $c>0$ (and say with the arbitrary and unecessary additional requirement that $c \neq 1$), there is a unique FLT $F \in \mathcal{S}$ such that the following three conditions hold:

$F(-1)=-1$, $F(0)=0$, and $\dfrac{F'(0)}{F'(-1)}=c^2$

I would like to ask for as many examples as possible of other classes of maps $\mathcal{C}$ (that is other than the class of FLT's) such that:

1. The same property as above of there existing a unique element in the class $\mathcal{C}$ satisfying the three conditions above holds.

2. Additionally, the class of maps $\mathcal{C}$ satisfies the property of being closed under (functional) composition of its members.

Thank you in advance to all those who respond,

E(up)lio M.