  We present infrared spectral energy distributions of 23 late-type T and Y
dwarfs obtained with the James Webb Space Telescope. The spectral energy
distributions consist of NIRSpec PRISM and MIRI LRS spectra covering the
$\sim$1--12 $\mu$m wavelength range at $\lambda/ \Delta \lambda \approx 100$
and broadband photometry at 15, 18, and 21 $\mu$m. The spectra exhibit
absorption features common to these objects including H$_2$O, CH$_4$, CO,
CO$_2$, and NH$_3$. Interestingly, while the spectral morphology changes
relatively smoothly with spectral type at $\lambda < 3$ $\mu$m and $\lambda >
8$ $\mu$m, it shows no clear trend in the 5 $\mu$m region where a large
fraction of the flux emerges. The broad wavelength coverage of the data enables
us to compute the first accurate measurements of the bolometric fluxes of cool
brown dwarfs. Combining these bolometric fluxes with parallaxes from Spitzer
and HST, we also obtain the first accurate bolometric luminosities of these
cool dwarfs. We then used the Sonora Bobcat solar metallicity evolutionary
models to estimate the radii of the dwarfs which results in effective
temperature estimates ranging from $\sim$1000 to 350 K with a median
uncertainty of $\pm$20 K which is nearly an order of magnitude improvement over
previous work. We also discuss how various portions of the spectra either do or
do not exhibit a clear sequence when ordered by their effective temperatures.
