  We introduce an abstract framework to study certain classes of stably
embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called
\textit{beautiful pairs}, which comprises Poizat's belles paires of stable
structures and van den Dries-Lewenberg's tame pairs of o-minimal structures.
Using an amalgamation construction, we relate several properties of beautiful
pairs with classical Fra\"iss\'e properties. After characterizing beautiful
pairs of various theories of ordered abelian groups and valued fields,
including the theories of algebraically, $p$-adically and real closed valued
fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of
henselian valued fields. As an application, we derive strict pro-definability
of particular classes of definable types. When $T$ is one of the theories of
valued fields mentioned above, the corresponding classes of types are related
to classical geometric spaces and our main result specializes to their strict
pro-definability. Most notably, we exhibit the strict pro-definability of a
natural space of types associated to Huber's analytifications. In this way, we
also recover a result of Hrushovski-Loeser on the strict pro-definability of
stably dominated types in algebraically closed valued fields, which corresponds
to Berkovich's analytifications.
