  Conformally soft gluons are conserved currents of the Celestial Conformal
Field Theory (CCFT) and generate a Kac-Moody algebra. We study celestial
amplitudes of Yang-Mills theory, which are Mellin transforms of gluon
amplitudes and take the double soft limit of a pair of gluons. In this manner
we construct the Sugawara energy-momentum tensor of the CCFT. We verify that
conformally soft gauge bosons are Virasoro primaries of the CCFT under the
Sugawara energy-momentum tensor. The Sugawara tensor though does not generate
the correct conformal transformations for hard states. In Einstein-Yang- Mills
(EYM) theory, we consider an alternative construction of the energy-momentum
tensor, similar to the double copy construction which relates gauge theory
amplitudes with gravity ones. This energy momentum tensor has the correct
properties to generate conformal transformations for both soft and hard states.
We extend this construction to supertranslations.
