  We present Ly$\alpha$ and ultraviolet-continuum (UV) luminosity functions
(LFs) of galaxies and active galactic nuclei (AGN) at $z=2.0-3.5$ determined by
the un-targetted optical spectroscopic survey of the Hobby-Eberly Telescope
Dark Energy Experiment (HETDEX). We combine deep Subaru imaging with HETDEX
spectra resulting in $11.4$ deg$^2$ of fiber-spectra sky coverage, obtaining
$18320$ galaxies spectroscopically identified with Ly$\alpha$ emission, $2126$
of which host type 1 AGN showing broad (FWHM$~>1000$ km s$^{-1}$) Ly$\alpha$
emission lines. We derive the Ly$\alpha$ (UV) LF over 2 orders of magnitude
covering bright galaxies and AGN in $\log
L_\mathrm{Ly\alpha}/\mathrm{[erg~s^{-1}]}=43.3-45.5$ ($-27<M_\mathrm{UV}<-20$)
by the $1/V_\mathrm{max}$ estimator. Our results reveal the bright-end hump of
the Ly$\alpha$ LF is composed of type 1 AGN. In conjunction with previous
spectroscopic results at the faint end, we measure a slope of the best-fit
Schechter function to be $\alpha_\mathrm{Sch}=-1.70^{+0.13}_{-0.14}$, which
indicates $\alpha_\mathrm{Sch}$ steepens from $z=2-3$ towards high redshift.
Our UV LF agrees well with previous AGN UV LFs, and extends to faint-AGN and
bright-galaxy regimes. The number fraction of Ly$\alpha$-emitting objects
($X_\mathrm{LAE}$) increases from $M_\mathrm{UV}^*\sim-21$ to bright magnitude
due to the contribution of type 1 AGN, while previous studies claim that
$X_\mathrm{Ly\alpha}$ decreases from faint magnitude to $M_\mathrm{UV}^*$,
suggesting a valley in the $X_\mathrm{Ly\alpha}-$magnitude relation at
$M_\mathrm{UV}^*$. Comparing our UV LF of type 1 AGN at $z=2-3$ with those at
$z=0$, we find that the number density of faint ($M_\mathrm{UV}>-21$) type 1
AGN increases from $z\sim2$ to $z\sim0$ as opposed to the evolution of bright
($M_\mathrm{UV}<-21$) type 1 AGN, suggesting the AGN downsizing in the
rest-frame UV luminosity.
