We present and explore deep narrow- and medium-band data obtained with the Subaru and the Isaac Newton Telescopes in the ~2deg^2^ COSMOS field. We use these data as an extremely wide, low-resolution (R~20-80) Integral Field Unit survey to slice through the COSMOS field and obtain a large sample of ~4000 Ly{alpha} emitters (LAEs) from z~2 to 6 in 16 redshift slices (SC4K). We present new Ly {alpha} luminosity functions (LFs) covering a comoving volume of ~10^8^Mpc^3^. SC4K extensively complements ultradeep surveys, jointly covering over 4dex in Ly{alpha} luminosity and revealing a global (2.5<z<6) synergy LF with {alpha}=-1.93^+0.12^_-0.12_, log_10_{Phi}^*^_Ly{alpha}_=-3.45^+0.22^_-0.29_Mpc^-3^, and log_10_L^*^_Ly{alpha}_=42.93^+0.15^_-0.11_erg/s. The Schechter component of the Ly{alpha} LF reveals a factor ~5 rise in L^*^_Ly{alpha}_ and a ~7 x decline in {Phi}^*^_Ly{alpha}_ from z~2 to 6. The data reveal an extra power-law (or Schechter) component above L_Ly{alpha}_~=10^43.3^erg/s at z~2.2-3.5 and we show that it is partially driven by X-ray and radio active galactic nucleus (AGN), as their Ly{alpha} 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 Ly {alpha} luminosity density rises by a factor ~2 from z~2 to 3 but is then found to be roughly constant (1.1^+0.2^_-0.2_x10^40^erg/s/Mpc^3^) to z~6, despite the ~0.7 dex drop in ultraviolet (UV) luminosity density. The Ly{alpha}/UV luminosity density ratio rises from 4+/-1 per cent to 30+/-6 per cent from z~2.2 to 6. Our results imply a rise of a factor of ~=2 in the global ionization efficiency ({xi}_ion_) and a factor ~=4+/-1 in the Ly{alpha} escape fraction from z~2 to 6, hinting for evolution in both the typical burstiness/stellar populations and even more so in the typical interstellar medium conditions allowing Ly{alpha} photons to escape.