Someone said :
One of my staff was emailed by the director this morning telling her he wants her to do a particular task for him tomorrow morning. She checked with me before replying to him to ask was that OK.
That's not how hierarchies work...
Actually, it is
She did check with you and you said no (or yes).
If she had felt it was an opportunity, she had the option to ask you to OK it, or to ask him to notify you of his orders/instructions.
It may not be how the policies in your department
are laid out as regard work assignments, ways of working, and how the hierarchy is specifically implemented.
It may not be how hierarchical courtesy
works: informing you would have been nice, and you were informed...
But in a hierarchy, the chain of command can most always be shortcut going down
: it is on the way up that shortcuts/bypasses are more restricted to basically grievances against the by-passed party. So it is how hierarchies work:
now you have the option to decline the services of your subordinate who informed you, and/or to discuss future ways of working "down the chain".
He was justified by the time-pressure / deadline to by-pass you. Especially if he has a pre-exiting positive working relationship with her.
If there is more to it than you exposed, you also have the option to express your grievance to his hierarchical superior![I assumed you worked for the Roinn Gnóthai Fostaíochta agus Coimirce Sóisialaí, but it could be the Army or any private company really]