The given coordination compound $[Ag(H_2O)_2][Ag(CN)_2]$ is composed of a complex cation $[Ag(H_2O)_2]^+$ and a complex anion $[Ag(CN)_2]^-$. Let the oxidation state of silver in both coordination spheres be $x$. Since the overall charge of the compound is zero, we can calculate the oxidation state as: $x + 2(0) + x + 2(-1) = 0 \Rightarrow 2x - 2 = 0 \Rightarrow x = +1$. Thus, the oxidation state of Ag is +1 in both the cationic and anionic entities.

• For the cationic part $[Ag(H_2O)_2]^+$, the ligands are neutral water molecules named 'aqua', and the metal is named 'silver'. So, it is named diaquasilver(I).
• For the anionic part $[Ag(CN)_2]^-$, the ligands are anionic cyanide ions named 'cyanido', and the metal name ends with '-ate', using its Latin root 'argentum'. So, it is named dicyanidoargentate(I). Combining the two parts (cation first, then anion), the full IUPAC name is diaquasilver(I) dicyanidoargentate(I).