  We study features of celestial CFT correlation functions when the bulk theory
is itself a CFT. We show that conformal inversions in the bulk map boost
eigenstates to shadow transformed boost eigenstates. This is demonstrated
explicitly for the wavefunctions of free massless scalars, and finds
interesting applications to building extrapolate dictionaries. Because
inversions exchange null infinity and the light cone of the origin, one finds a
relation between the massless extrapolate dictionary -- involving correlators
of operators inserted along null infinity -- and the slice-by-slice extrapolate
dictionary recently studied by Sleight and Taronna starting from the hyperbolic
foliation of de Boer and Solodukhin. Namely, boundary correlators of Sleight
and Taronna coincide with celestial amplitudes of shadow transformed boost
eigenstates. These considerations are unified by lifting celestial correlators
to the Einstein cylinder. This also sheds new light on the extraction of the
$S$-matrix from the flat limit of AdS/CFT.
