  We extend Shen's recent formulation (arXiv:1806.07388) of the classical
double copy, based on explicit color-kinematic duality, to the case of
finite-size sources with non-zero spin. For the case of spinning Yang-Mills
sources, the most general consistent double copy consists of gravitating
objects which carry pairs of spin degrees of freedom. We find that the
couplings of such objects to background fields match those of a classical (i.e.
heavy) closed bosonic string, suggesting a string theory interpretation of
sources related by color-kinematics duality. As a special case, we identify a
limit, corresponding to unoriented strings, in which the 2-form Kalb-Ramond
axion field decouples from the gravitational side of the double copy. Finally,
we apply the classical double copy to extended objects, described by the
addition of finite-size operators to the worldline effective theory. We find
that consistency of the color-to-kinematics map requires that the Wilson
coefficients of tidal operators obey certain relations, indicating that the
extended gravitating objects generated by the double copy of Yang-Mills are not
completely generic.
