Tenant wants to back out from moving in

[Jack B.]

Their current landlord will allow them to stay after all due to his own housing situation. They gave me a 2K deposit a month ago. I have nothing in writing other than our emails about the lease and moving in, but he called me today he wants to back out and stay where he is at. He wants part of his deposit back if I find a new tenant in time. He did just say via text that he understands if I don't find a replacement in time, but wonders about if I do.

I told him no, and that it's unlikely that would happen as it took me 2 months to find him. Now I have less than a month to find a replacement for a class A property....

Thoughts?

ETA: I will go by the book for every little receipt with tenants from now on.