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

This article references checking the actual code file, but to make it easier you can also look at the _satellite object in the Console if your web browser.

_satellite.buildDate = the last build date of either the production or staging library
_satellite.publishDate = the last date the production library was last published

These might be beneficial to pass into a prop or an eVar to see the effects of caching with the data or other troubleshooting purposes.

I have to agree with Adam here.  Using the functions below is the fastest and most reliable. 

_satellite.buildDate = the last build date of either the production or staging library
_satellite.publishDate = the last date the production library was last published