There is mounting evidence that, in some sense, gravity can be seen as a double copy of two gauge theories. In this talk I will discuss how this structure emerges for certain amplitudes containing soft gravitons. We will see how the leading and next-to-leading terms of the soft-graviton theorem, when written using kinematic invariants, exhibit a double copy structure for the emitted graviton. This result holds also in the Gribov limit, where gravitons are not necessarily soft.