It’s not as hard as it looks if we ground our apology in facts.
We say the following:
It is a fact that this happened.
I understand that this happened.
I apologize to you for my role in making this happen / not stopping it from happening.
I feel genuine remorse about the fact that it happened.
Owning our mistakes is one of the bravest, most empathetic things we can do. No fooling.
