The Sec of War can relay the orders of the CinC in his name. That's common with any military Chief of Staff position. You really don't know how military chains of command work.
When I had to submit memorandums and reports on stuff I was tasked with doing, I would sign them and add the line "By direction." That meant that the CO or other officer that was ultimately in charge had given me an order / permission to do the memo or report and that I didn't need to have them sign it. I signed it in their name using that tag.
Same thing here. Trump gave the orders, Hegseth carried them out "By direction," to use the Navy terminology. The other services may vary some from that, but each has its proper bureaucratic etiquette you follow.