  Let $X$ be a projective K3 surfaces. In two examples where there exists a
fine moduli space $M$ of stable vector bundles on $X$, isomorphic to a Hilbert
scheme of points, we prove that the universal family $\mathcal{E}$ on $X\times
M$ can be understood as a complete flat family of stable vector bundles on $M$
parametrized by $X$, which identifies $X$ with a smooth connected component of
some moduli space of stable sheaves on $M$.
