  The characteristic Cauchy problem of the Einstein field equations has been
recently addressed from a completely abstract viewpoint by means of
hypersurface data and, in particular, via the notion of double null data.
However, this definition was given in a partially gauge-fixed form. In this
paper we generalize the notion of double null data in a fully diffeomorphism
and gauge covariant way, and show that the definition is complete by proving
that no extra conditions are needed to embed the double null data in some
spacetime. The second aim of the paper is to show that the characteristic
Cauchy problem satisfies a geometric uniqueness property. Specifically, we
introduce a natural notion of isometry at the abstract level such that two
double null data that are isometric in this sense give rise to isometric
spacetimes.
