  We show that two- and three-point celestial (C)CFT$_{d-1}$ amplitudes can be
directly obtained from correlation functions in a unitary Lorentzian CFT$_d$ on
$\mathbb{R}\times S^{d-1}$. The recipe involves a rescaling of the operators,
followed by an expansion around a bulk point configuration and a transformation
to an $S^{d-1}$ conformal primary basis. The first two steps project the
CFT$_d$ correlators onto distributions on $S^{d-1}$. The final step implements
a dimensional reduction yielding CCFT$_{d-1}$ amplitudes that are manifestly
vanishing for all in/out configurations and Poincar\'e invariant. The
dimensional reduction may be implemented either by evaluating certain time
integral transforms around the bulk-point limit, or by analytically continuing
the CFT$_d$ operator dimensions and restricting the operators to $S^{d-1}$ time
slices separated by $\pi$ in global time. The latter prescription generates the
correct normalization for both two- and three-point functions. On the other
hand, the celestial three-point amplitudes obtained via the former prescription
are found to only agree after evaluating a residue at an integer linear
combination of the CFT$_d$ conformal dimensions. The correct normalization may
also be obtained by considering a different integration path in the uplift of
the complexified time plane to its universal cover.
