The Kontakt Factory Library Path in the Load / Import tab of the KONTAKT 4 Main Options contains the path where the KONTAKT Library is stored. If the entered path is not correct, the Factory Library will not appear in the KONTAKT Library browser. You can define the Library Path by clicking the Choose button and navigating to the folder where the Factory Library is stored.
The folders specified in the Database tab are used to allow the KONTAKT 4 database for browsing KONTAKT patches by categories or search for certain files. The Database should also contain the Factory Library path. It is added to the database path(s) automatically if you install KONTAKT 4 with the library component selected. However, if you install KONTAKT 4 without the library component (e.g. since you already have the library on an external hard disk), then you need to manually Add your KONTAKT 4 library path to the Database tab and press the Update button afterwards to import the library to the database.