I published, how do I check if my library updated?

At the bottom of the library file, there's a notation of when the file was last updated. It looks something like the following:

buildDate:"2014-11-03 02:47:04 UTC",publishDate:"2014-10-29 21:45:37 UTC"

One way to open your library file is to go to the 'Embed' tab in the DTM property, expand the 'Header Code' section, and copy the production code. Then, paste the copied text into the browser and remove the script tags and add 'http:'

So, the copied embed '<script src="//assets.adobedtm.com/123456abcdefg/satelliteLib-fc37eab6859bb77b90e6a30892b9.js"></script>

Will become 'http://assets.adobedtm.com/123456abcdefg/satelliteLib-fc37eab6859bb77b90e6a30892b9.js'

This link reveals the production library file and the publish / build date is reflected at the bottom. Compare this to your last publish date from the property history tab to check if the file has updated.

It is also recommended to test, and ensure that the changes are working as expected on your site by testing in the console. It is a best practice to test in production after a publish.