Mouse receiver stopped working after pairing mouse to an unifying receiver

At the end, after asking the seller for a solution to my problem, he sent me the following link: http://support.logitech.com/software/logitech-connection-utility-software

It's the software Logitech Connection Utility, copy pasted from the link I put above: "It can be used to reconnect supported keyboards or mice with a replacement receiver or when the connection is lost."

It solved instantly the problem I described in my question.