You can use your own mug, but they charge per the size of the mug. So I believe it ends up being the same price as the cup sizes... if your mug is large, they charge for a 'large' coffee, etc. I don't think there is a discount.
HOWEVER. If you buy coffee from Union Market.... then you get a deal if you use your mug
Did you know that coffee cups are not recyclable? Not even the lids are recyclable on campus (although they are recyclable in the City of Hamilton). Campus goes through tens of thousands of coffee cups every year! To reduce waste and get a deal, pick up a reusable travel mug! If you use your mug at Union Market, you'll only get charged for a medium no matter how big your mug is!