  Inspired by the theory of classifying topoi for geometric theories, we define
rounded sketches and logoi and provide the notion of classifying logos for a
rounded sketch. Rounded sketches can be used to axiomatise all the known
fragments of infinitary first order logic in $\mathbf{L}_{\infty,\infty}$, in a
spectrum ranging from weaker than finitary algebraic to stronger than
$\lambda$-geometric for $\lambda$ a regular cardinal. We show that every
rounded sketch has an associated classifying logos, having similar properties
to the classifying topos of a geometric theory. This amounts to a
Diaconescu-type result for rounded sketches and (Morita small) logoi, which
generalises the one for classifying topoi.
