Spin vector and operator transformations query

[deneve]

I am struggling to understand spin transformations and have used Sakurai's method of
|new basis> = U |old basis> to change basis vectors and hence should have

Sz' = Udagger Sz U

to transform the operator. I thought this should give Sz' = Sy in the workings (see attachment below) but it gives Sx instead. If I use
Sz' = U Sz Udagger then I do get Sz'=Sy but according to the texts I have, the first arrangement is the correct form. So I'm really confused. Can anyone see what I'm doing wrong here.


2. Homework Equations

I would be really grateful for any help with this. Thank you

 

[azntoon]

. 3. The Attempt at a Solution (The attachments are not letting me paste the math here so I have posted the link to my calculation instead). My Calculation: https://ibb.co/n6nC9x