  We formulate a new program to generalize the double-copy of tree amplitudes.
The approach exploits the link between the identity element of the KLT algebra
and the KLT kernel, and we demonstrate how this leads to a set of KLT bootstrap
equations that the double-copy kernel has to satisfy (in addition to locality
constraints). We solve the KLT bootstrap equations perturbatively to find the
most general higher-derivative corrections to the 4- and 5-point field theory
KLT kernel. The new kernel generalizes the string KLT kernel and its associated
monodromy relations. It admits new color-structures in the effective theories
it double-copies. It provides distinct generalized KK and BCJ relations for the
left and right single-color theories and is in that sense a heterotic-type
double-copy. We illustrate the generalized double-copy in detail for 4d
Yang-Mills theory with higher-derivative corrections that produce
dilaton-axion-gravity with local operators up order $\nabla^{10} R^4$. Finally,
we initiate a search for new double-copy kernels.
