@wesen99 the way I would do it (other might not agree) is to have 4 folders in the Documents library on the site, one for each team. I wouldn't do it with 4 different documents libraries as that makes it harder to combine the final documents into a single list for the homepage later.
You would set the permissions for each folder and therefore for all the documents in it by clicking on the 3 dots next to the folder name in the top level of the Documents Library and selecting Manage Access.
In the Documents Library I would then have a custom column to hold the project name. This would appear in each folder so you only need to do it at the top level. The documents for each project for each team could be uploaded to the relevant folder by each team leader and the project name set so that s/he could filter the library for a specific project when necessary.
I would also have a Status column so that you could set each document status to Final when you wanted to.
Next, create a view of the Documents Library and make sure you set the Folders section to "Show all items without folders". Set the filter to Status is equal to Final.
To display all the final documents on the homepage, add a Document Library web part, select the Documents Library and the view you created above: