You can do this with azcopy sync. You can synchronize the contents of a file share with another file share and you can also synchronize the contents of a directory in a file share with the contents of a directory that is located in another file share. Synchronization is one way. In other words, you choose which of these two endpoints is the source and which one is the destination. Synchronization also uses server to server APIs. I believe this specific example is what you are trying to achieve.
Hope this helps! Please try this and see if it works for you. Let me know if you have any questions or need further assistance.
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