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Let M be a 2n-dimensional smooth and compact moduli space of stable sheaveson a K3 surface S and U a universal sheaf over S x M. Over M x M there exists anatural reflexive sheaf E of rank 2n-2, namely the first relative extensionsheaf of the two pullbacks of U to M x S x M. We prove that E is slope-stablewith respect to every Kahler class on M. The sheaf E is known to deform to asheaf E' over X x X, for every manifold X deformation equivalent to M, and weprove that E' is slope-stable with respect to every Kahler class on X. Thistriviality of the stability chamber structure combines with a result of S.Mehrotra and the author to show that the deformed sheaf E' is canonical.Consequently, the pretriangulated K3 category associated to the pair (X x X,E')in our earlier work with S. Mehrotra depends only on the isomorphism class ofX.

Naturality of the hyperholomorphic sheaf over the cartesian square of a manifold of $K3^{[n]}$-type