  We explore deep rest-frame UV to FIR data in the COSMOS field to measure the
individual spectral energy distributions (SED) of the ~4000 SC4K (Sobral et al.
2018) Lyman-alpha (Lya) emitters (LAEs) at z~2-6. We find typical stellar
masses of 10$^{9.3\pm0.6}$ M$_{\odot}$ and star formation rates (SFR) of
SFR$_{SED}=4.4^{+10.5}_{-2.4}$ M$_{\odot}$/yr and SFR$_{Lya}=5.9^{+6.3}_{-2.6}$
M$_{\odot}$/yr, combined with very blue UV slopes of beta=-2.1$^{+0.5}_{-0.4}$,
but with significant variations within the population. M$_{UV}$ and beta are
correlated in a similar way to UV-selected sources, but LAEs are consistently
bluer. This suggests that LAEs are the youngest and/or most dust-poor subset of
the UV-selected population. We also study the Lya rest-frame equivalent width
(EW$_0$) and find 45 "extreme" LAEs with EW$_0>240$ A (3 $\sigma$), implying a
low number density of $(7\pm1)\times10^{-7}$ Mpc$^{-3}$. Overall, we measure
little to no evolution of the Lya EW$_0$ and scale length parameter ($w_0$)
which are consistently high (EW$_0=140^{+280}_{-70}$ A, $w_0=129^{+11}_{-11}$
A) from z~6 to z~2 and below. However, $w_0$ is anti-correlated with M$_{UV}$
and stellar mass. Our results imply that sources selected as LAEs have a high
Lya escape fraction (f$_{esc, Lya}$) irrespective of cosmic time, but f$_{esc,
Lya}$ is still higher for UV-fainter and lower mass LAEs. The least massive
LAEs ($<10^{9.5}$ M$_{\odot}$) are typically located above the star formation
"Main Sequence" (MS), but the offset from the MS decreases towards z~6 and
towards $10^{10}$ M$_{\odot}$. Our results imply a lack of evolution in the
properties of LAEs across time and reveals the increasing overlap in properties
of LAEs and UV-continuum selected galaxies as typical star-forming galaxies at
high redshift effectively become LAEs.
