  For a compact group $\mathbb{G}$, the functor from unital Banach algebras
with contractive morphisms to metric spaces with 1-Lipschitz maps sending a
Banach algebra $A$ to the space of $\mathbb{G}$-representations in $A$
preserves filtered colimits. Along with this, we prove a number of analogues:
one can substitute unitary representations in $C^*$-algebras, as well as
semisimple finite-dimensional Banach algebras (or finite-dimensional
$C^*$-algebras) for $\mathbb{G}$.
  These all mimic results on the metric-enriched finite
generation/presentability of finite-dimensional Banach spaces due to Ad{\'a}mek
and Rosick{\'y}. We also give an alternative proof of that finite
presentability result, along with parallel results on functors represented by
compact metric, metric convex, or metric absolutely convex spaces.
