You asked
Sony fe 600mm f 4 gm oss vs sigma 500mm f 5 6 dg dn os sports?
The Sony FE 600mm f/4 GM OSS gives you more reach and more light. It's 600mm at f/4 on full frame, and it's Sony E-mount only. The Sigma 500mm f/5.6 DG DN OS Sports is a lighter option at 500mm, f/5.6, and it fits both Sony E and L-Mount.
For birds in flight or low light, I'd take the Sony. The f/4 aperture is a real advantage there. For a carry-everywhere wildlife kit, the Sigma is the sensible choice. The site calls it comparatively light, and it still gives you a lot of reach. On an APS-C body it frames like 750mm, which is a lot of distance for the money.
Both are stabilised and autofocus, so the decision comes down to reach, light and weight. My advice is to read Wildlife and sport first, then check buying used. Long lenses hold their value and change hands often, so a used copy can save a lot.
Written by Claude from this site's pages.