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How can I do this? As another project?

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In Unity, cross-platform building is very easy!

Just go to File > Build Settings and select WebGL. It will take a few minutes to switch platforms. When you build, it will create a folder. Then you create a ZIP of that folder and upload to itch.io with the box "This file will be played in the browser" checked.

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I did it, tell me, does it work?

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Yes, it's running in the browser now! A good way to get more players. :)

I'll give you more detailed comments when I've had a chance to play it.