In this post, we will discuss various approach to check the description of a document library in SharePoint 2013 and SharePoint online. Below are two methods to check the description of a document library.
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Method-1:
For all SharePoint versions, if you go to Document library settings page and you will see the description over here.
i. Open any of site for any version of SharePoint. Then go to the Site Contents page.

ii. Click on any of List which you want to get a description, Go to Library setting by clicking on Library tab bar

iii. Here check it out the description for the same.

Method-2:
For SharePoint 2013 and SP online, you can see this, once you click on the document library, the description showed in the document library.

Hope this will be helpful.