We prove a diagonalisation theorem for the tautological, or generalised Miller--Morita--Mumford, classes of compact, smooth, simply connected, definite $4$--manifolds. Our result can be thought of as a families version of Donaldson's diagonalisation theorem. We prove our result using a families version of the Bauer--Furuta cohomotopy refinement of Seiberg--Witten theory. We use our main result to deduce various results concerning the tautological classes of such $4$--manifolds. In particular, we completely determine the tautological rings of $mathbb{CP}^2$ and $mathbb{CP}^2 mathbin{#} mathbb{CP}^2$. We also derive a series of linear relations in the tautological ring which are universal in the sense that they hold for all compact, smooth, simply connected definite $4$--manifolds.
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