complex analysis – holomorphic functions reflected by segments that are not on the real axis

My question concerns the use of the Schwarz Reflection principle (or principle of symmetry) to reflect the regions of the domain of a holomorphic function in a symmetric region (with respect to a straight line segment) in which $ f (z) $ is already defined.

More specifically, if $ f: mathbb {D} rightarrow mathbb {C} $ is a holomorphic function that has a real value along the line $ arg z = pi / 3 $we can take a "side" of a region whose boundary is along $ arg z = pi / 3 $ and think through it via the formula $ tilde {f} ( tilde {z}) $, where ~ denotes the reflection. As it is an analytical continuation and $ f (z) $ is already defined on the other "side" of $ arg z = pi / 3 $, does it make sense to conclude that $ f (z) = f ( tilde z) $?

Moreover, when the line segment in question is the real axis, the map is actually the usual complex conjugation. Initially, I wanted to tackle this issue by turning $ arg z = pi / 3 $ clockwise $ pi / $ 3 to bring the real value segment back to the axis, apply the principle of symmetry here and rotate backwards. Unless I'm mistaken, I think it shows that $ tilde f ( tilde z) = overline {f ( overline {z})} $ ? That is, reflection by line segments that are not on the real axis always follows the same reflection formula. It seems wrong. I would like to have a glimpse of my two (apparently fake) observations .. thanks!