Honestly, finding a lens that works well on both full frame and crop sensor bodies is kinda annoying. I tried using the Sony FE 50mm f/1.8 on my a6000 and the autofocus was honestly pretty disappointing. It hunts way too much and feels sluggish compared to newer glass, which sucks when you are just trying to grab a quick street shot. If you really need something that plays nice with both, maybe check these out instead:

• Sony FE 28mm f/2.0 is way snappier. It feels pretty wide on the a7iii and acts like a nice 42mm equivalent on the a6000.
• Samyang AF 35mm f/2.8 FE is tiny. It is not as sharp as the more expensive stuff, but for the price, it is basically a pancake lens that makes your street setup super portable. Dont expect miracles tho, the autofocus on the older bodies like the a6000 really shows its age with these budget primes.

Quick reply while I have a sec! If you want a sharp lens that is actually fun to use on both bodies, you gotta check out the Samyang AF 35mm f/2.8 FE. It is ridiculously tiny and snappy for street stuff.

• Super light for your a6000
• Great focal length for full frame Are you looking for a prime or a zoom? Knowing if you need something wider or longer would help me narrow it down!

Jumping in here cause I learned this the hard way tbh. Trying to use full frame lenses on an a6000 for street stuff sounds good in theory, but honestly, weight distribution gets weird fast. I bought a Tamron 28-75mm f/2.8 Di III VXD G2 thinking it would be my one-lens solution for everything, and unfortunately, it just felt way too front-heavy on the smaller body. My wrist was killing me after two hours of walking. TL;DR: Keep your expectations low for cheap full frame glass on crop bodies. Quick tip: If you want something safe that wont break your budget or your wrist, grab a dedicated cheap prime for the a6000 instead of forcing a crossover lens. Your arms will thank you later.