Is showing off your girl empowering or objectifying

A topic came up in class that seemed controversial to me. How if you were to show off your girlfriend would be objectifying her, and that is messed up. The argument is your showing off your girlfriend like you would be showing off a Ferrari, and she is therefore no more than an object to you. However, I know that some girls actually like to be shown off by their boyfriends to their friends because they feel like they are special to the man. Is showing someone off necessarily objectifying? Can you show off your girlfriend like a little kid would show off their dad? Or is it bad to show a selfie of your girlfriend to your friend and say 'aren't I lucky? ' Maybe you should ask your girlfriend beforehand.  And is it objectifying me when my girl shirtless pics of me to her friends without asking me? Tell me why I like it though.