  A simple model for the localization of the category $\mathbf{CLoc}_2$ of
oriented and time-oriented globally hyperbolic conformal Lorentzian
$2$-manifolds at all Cauchy morphisms is constructed. This provides an
equivalent description of $2$-dimensional conformal algebraic quantum field
theories (AQFTs) satisfying the time-slice axiom in terms of only two algebras,
one for the $2$-dimensional Minkowski spacetime and one for the flat cylinder,
together with a suitable action of two copies of the orientation preserving
embeddings of oriented $1$-manifolds. The latter result is used to construct
adjunctions between the categories of $2$-dimensional and chiral conformal
AQFTs whose right adjoints formalize and generalize Rehren's chiral
observables.
