This is compassion / responsiveness, not empathy.

Empathy = mental simulating of how an action will affect another.

Compassion = a desire to help another.

Responsiveness = the ability to recognize changes in someone's emotional state, and then develop and engage in a course of action based upon that change.

More info: Heinz Kohut - Reflections on Empathy https://www.youtube.com/watch?v=ZQ6Y3hoKI8U