[A Proposition from a book written by Benson Farb and Dan Margalit](https://i.stack.imgur.com/dbTFZ.png)



How to understand this isomorphism?
The question is terribly put , but the answer is: $S\_{0, 4}$ is the four times punctured sphere. You can think of this sphere as the ideal simplex in $\mathbb{H}^3$ (it is a theorem of mine that this is always possible). A simplex always has a Klein four-group worth of symmetries, hence the $\mathbb{Z}/2 \mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ factor. The quotient is $S\_{0, 2, 2, 2},$ which is the quotient of a punctured torus by the elliptic involution. The mapping class group of a punctured torus is $PSL(2, \mathbb{Z})$ and every element commutes with the elliptic action.


