  Recent work by Cachazo, He, and Yuan shows that connected prescription
residues obey the global identities of $\mathcal{N} = 4$ super-Yang-Mills
amplitudes. In particular, they obey the Bern-Carrasco-Johansson (BCJ)
amplitude identities. Here we offer a new way of interpreting this result via
objects that we call residue numerators. These objects behave like the
kinematic numerators introduced by BCJ except that they are associated with
individual residues. In particular, these new objects satisfy a double-copy
formula relating them to the residues appearing in recently-discovered analogs
of the connected prescription integrals for $\mathcal{N} = 8$ supergravity.
Along the way, we show that the BCJ amplitude identities are equivalent to the
consistency condition that allows kinematic numerators to be expressed as
amplitudes using a generalized inverse.
