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First, go to your Google Tag Manager screen and choose the appropriate project and container.
Start by "Add Tag". Create a Custom HTML Tag, call it "Thinglink" and copy the code found here into the "HTML" section.
Don't forget to set up a rule (such as "All pages") or Thinglink won't be available. It's safe to put Thinglink on every page, since we don't interfere with the operation of the page if you're not entitled to edit any images, or if the images have no tags.
Save the tag, then publish. In a few moments you should see Thinglink edit button appearing on all eligible images on your web site!
(Note that the edit button is only visible if you actually have edit rights on the images; your casual visitor will not see the edit button at all.)