When the temperature of a source and a sink becomes equal, the entropy change is determined by the process that leads to the equalization of temperatures.
If the equalization occurs through a reversible process, then the entropy change is zero. In a reversible process, the system goes through a series of equilibrium states, and entropy is conserved. When the temperatures of the source and sink become equal in a reversible process, the entropy change of the system is zero.
However, if the equalization occurs through an irreversible process, then the entropy change is typically positive. In an irreversible process, the system does not pass through a series of equilibrium states, and entropy is generated. When the temperatures of the source and sink become equal in an irreversible process, the entropy of the system increases.
In summary, if the equalization of temperatures occurs through a reversible process, the entropy change is zero. If it occurs through an irreversible process, the entropy change is positive.