The reaction between Cu2+ and KI will takes place as:
Cu2++2KI⟶UnstableCuI2↓+2K+
I− is a strong reducing agent, it reduces Cu2+ to Cu+ and a white precipitate of Cu2I2 is formed.
2CuI2⟶(White)′X′Cu2I2↓+I2
Sodium thiosulphate is used as reducing agent to titrate iodine, with iodide being the product of the reaction.
2Na2S2O3+I2→2NaI+(Y)Na2S4O6
So, the compound XandY are Cu2I2andNa2S4O6 respectively.