In field theory, the "double copy" can be thought of as a multiplicative map between tree amplitudes of two field theories into tree amplitudes of another theory. The origin of the map is string theory, but from a bottom-up Effective Field Theory approach, a natural question is if there exists other versions of the double-copy map. In the talk I will review the double copy and a novel recent approach to generalize it. I will discuss some new algebraic structures that this "bootstrap" of the double-copy entails.