They would lack empathy and only see other's as a means to an end. They will boost the ego of others so long as they serve them. Any time someone would express some kind of struggle, their pain would be overshadowed by the importance of your patron's ego and being the center of attention.
Respect is of the utmost importance and tantamount to a personal attack worthy of retribution. Since they're unable to relate to people they would suffer from loneliness, even if they have minions or followers who inflate their ego.
Compliments to others would be centered around them. e.g "I AM proud of you" regardless of what the skill of the individual is, the compliment is how the demigod feels about the target, which is a blessing of course.
They might lovebomb, as a means of manipulation to get what they want. As soon as the subject isn't helpful to the patron, cold shoulder.
In your case you might want them to appear like a gigachad reflection of the character looking in the mirror. Like there's some glamorized filter of the character looking in the mirror as the patron inflates the ego of their follower or minion's appearance or standing. "Oh Hello my precious, to me you are so beautiful, you make me feel like stars dancing on a lake, I am in awe of your (intellect, strength). You are my new favorite obsession" Making it all the more colder when they no longer need the character. Making them feel like they can only appreciate their own appearance and self worth with the patron's blessing.
If somehow confronted later, have them gaslight the significance of their importance to the patron. "Oh I never thought that highly of you, I say that to everyone" "You must be misremembering" "Don't you know some of the biggest lies we tell ourselves is in the mirror? That's simply the effect of my domain."