This article explains what the "inventory sync" option does automatically in the background if enabled - and what other tasks require manual interaction or can be automated on their own (like requesting and processing an inventory report).
How "Inventory Sync" works
With inventory sync enabled, WP-Lister will sync your sales on Amazon and WooCommerce: when an item sells on Amazon, the stock is reduced in WooCommerce - and vice versa: When an item sells in WooCommerce, it's stock is reduced on Amazon.
However, if you manually change the stock levels on Amazon, these will not be synced back to WooCommerce automatically - at least not by enabling the inventory sync.
Why WooCommerce becomes your central database
When using WP-Lister for Amazon you should not be making manual changes on Amazon anymore. Instead, your changes should be made directly in WooCommerce - and WP-Lister will take care of uploading your changes to Amazon.
One of WP-Lister's core features is submitting products and changes from WooCommerce to Amazon. This is what WP-Lister does "out of the box" - and it's not required to enable the inventory sync option to do so. When you update a product in WooCommerce, all the data that has already been sent to Amazon will be submitted again with your update. That's why when you use WP-Lister, WooCommerce will become your "central database".
Another option - process a daily inventory report
That being said, if you have to submit inventory changes to Amazon directly (for example because you have some 3rd party system in place which does that automatically) and need these to be reflected in WooCommerce, there is a simple way to do so: All you need to do is to request a new inventory report and let WP-Lister process it - with the import option to update stock levels enabled.
There is an option to request and process an inventory report on a daily basis on Amazon -> Settings -> Advanced. This will update prices and/or stock levels (if enabled in the report processing options), but it will not import new products automatically that you might have added on Amazon. And your changes would be applied more quickly if you request and process the report manually.
Note: An inventory report takes some time to be generated, and there is a limit on how many reports of each type you can request per day.