This is basically a small high horn used to play some baroque and later pieces - see for example https://de.wikipedia.org/wiki/Corno_da_caccia (in German, but you can see what it is). Unlike a modern French horn they are played with the right hand operating the valves and the bell facing outwards.

There is a lot of conjecture over what instrument composers intended when they specified corno da caccia, but since this is about what instruments are actually used in recorded performances, and these instruments exist in this form and are used, I don't think that matters.