S map is a function which maps "a pair of a natural number and a function" to "a pair of a natural number and a function". It was defined by Japanese googologist Fish in 2002 and used in the definition of Fish number 1 and Fish number 2. It is defined as \begin{eqnarray*} S:[m,f(x)]→[g(m),g(x)] \end{eqnarray*} which means that when a pair of \(m \in \mathbb{N}\) and a function \(f(x)\) is given as input variables of S map, a pair of \(g(m) \in \mathbb{N}\) and a function \(g(x)\) is obtained as return values, where \(g(x)\) is defined as and \(g(m)\) is calculated by substituding \(x=m\) to \(g(m)\).