In 2017, Duchi, Guerrini, Rinaldi and Schaeffer proposed a new combinatorial object called "fighting fish", which is counted by the same formula as more classical objects, such as two-stack sortable permutations and non-separable planar maps. In this article, we explore the bijective aspect of fighting fish by establishing a bijection to two-stack sortable permutations, using a new recursive decomposition of these permutations. With our bijection, we give combinatorial explanations of several results of fighting fish previously proved previously with generating functions. Using the decomposition, we also prove the algebraicity of a generating function of two-stack sortable permutations, extending a result of Bousquet-M\'elou (1998). 查看全文>>