오늘 제가 대학로에 연극을 보러 갈 거예요.
I read this sentence in my textbook. I remember 제 should mean "my". But in this sentence, it seems like 제 can also mean "I". Is that true? Thank you very much.

Question

Answer

Yes. '제' means 'I' in your sentence. 
'제' is an honorific (humble) form of 'I' which means you lower yourself to show your respect to the listener.

'저' is changed to '제' with the subject marker '가' while you say '저' with the topic marker '는'.

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