  We provide a universal characterization of the construction taking a scheme
$X$ to its stable $\infty$-category $\text{Mot}(X)$ of noncommutative motives,
patterned after the universal characterization of algebraic K-theory due to
Blumberg--Gepner--Tabuada. As a consequence, we obtain a corepresentability
theorem for secondary K-theory. We envision this as a fundamental tool for the
construction of trace maps from secondary K-theory.
  Towards these main goals, we introduce a preliminary formalism of "stable
$(\infty, 2)$-categories"; notable examples of these include (quasicoherent or
constructible) sheaves of stable $\infty$-categories. We also develop the
rudiments of a theory of presentable enriched $\infty$-categories -- and in
particular, a theory of presentable $(\infty, n)$-categories -- which may be of
intependent interest.
