  In 2017, Duchi, Guerrini, Rinaldi and Schaeffer proposed a new family of
combinatorial objects called "fighting fish", which are 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 on fighting fish proved
previously with generating functions. Using the decomposition of two-stack
sortable permutations, we also prove the algebraicity of their generating
function, extending a result of Bousquet-M\'elou (1998).
