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 $log L_mathrm{Lyalpha}/mathrm{[erg~s^{-1}]}=43.3-45.5$ ($-27<M_mathrm{UV}<-20$) 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 $X_mathrm{Lyalpha}$ decreases from faint magnitude to $M_mathrm{UV}^*$, suggesting a valley in the $X_mathrm{Lyalpha}-$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 $zsim2$ to $zsim0$ as opposed to the evolution of bright ($M_mathrm{UV}<-21$) type 1 AGN, suggesting the AGN downsizing in the rest-frame UV luminosity.
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