  We present and explore deep narrow- and medium-band data obtained with the
Subaru and the Isaac Newton telescopes in the ~2 deg$^2$ COSMOS field. We use
these data as an extremely wide, low-resolution (R~20-80) IFU survey to slice
through the COSMOS field and obtain a large sample of ~4000 Lyman-$\alpha$
(Lya) emitters from z~2 to z~6 in 16 redshift slices (SC4K). We present new Lya
luminosity functions (LFs) covering a co-moving volume of ~10$^8$Mpc$^3$. SC4K
extensively complements ultra-deep surveys, jointly covering over 4 dex in Lya
luminosity and revealing a global (2.5<z<6) synergy LF with
$\alpha=-1.93\pm0.12$, $\log_{10}\Phi^*=-3.45^{+0.22}_{-0.29}$ Mpc$^{-3}$ and
$\log_{10}L^*=42.93^{+0.15}_{-0.11}$ erg/s. The Schechter component of the Lya
LF reveals a factor ~5 rise in $L^*$ and a ~7x decline in $\Phi^*$ from z~2 to
z~6. The data reveal an extra power-law (or Schechter) component above
L~10$^{43.3}$ erg/s at z~2.2-3.5 and we show that it is partially driven by
X-ray and radio AGN, as their Lya LF resembles the excess. The power-law
component vanishes and/or is below our detection limits above z>3.5, likely
linked with the evolution of the AGN population. The Lya luminosity density
rises by a factor ~2 from z~2 to z~3 but is then found to be roughly constant
(~$1.1\times10^{40}$ erg s$^{-1}$ Mpc$^{-3}$) to z~6, despite the ~0.7 dex drop
in UV luminosity density. The Lya/UV luminosity density ratio rises from
$4\pm1$% to $30\pm6$% from z~2.2 to z~6. Our results imply a rise of a factor
of ~2 in the global ionisation efficiency ($\xi_{\rm ion}$) and a factor
~$4\pm1$ in the Lya escape fraction from z~2 to z~6, hinting for evolution in
both the typical burstiness/stellar populations and even more so in the typical
ISM conditions allowing Ly$\alpha$ photons to escape.
