You can bring it, but they will not give you money off. I've seen people bring coolers into Typhoon Lagoon, they are just subject to search.
Forget the cup. Bring a cooler. Just can't bring in alcohol or glass
I know you can bring Disney resort mugs and pay to have them activated to use at TL which is about $4 cheaper than buying their souvenir cup not sure if you can do the same for a Universal mug
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