The Ludicrous Labour

[Angrboda]

Trans woman: male to female.
Trans man: female to male.

So, a "trans woman" is someone of the male sex (a man) who "transitioned" to the female sex gender? 

And a "trans man" is someone of the female sex (a woman) who "transitioned" to the male sex gender? 

But how do they do that? I mean, even if they have an operation, it doesn't change their sex, it only changes their appearance. It's still a man, a male. The operation changes neither their sex, nor what they think they are.

FTFY. You're basically begging the question if you use male sex as a synonym for man, and similarly with female sex. It's implicitly rejecting the concept of gender.