A round disc of moment of inertia $I_2$ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia $I_1$ rotating with an angular velocity $\omega$ about the same axis. The final angular velocity of the combination of discs is :-

No external torque acts on the system of two discs (the friction between them is internal). So angular momentum is conserved.
Initial angular momentum $= I_1\omega$ (the second disc is initially at rest).
Final angular momentum $= (I_1 + I_2)\omega'$, since both discs finally rotate together.
From conservation of angular momentum: $I_1\omega = (I_1 + I_2)\omega'$
$\Rightarrow \omega' = \dfrac{I_1\omega}{I_1 + I_2}$