Remember that the velocity of the block has no vertical component at all times as gravity is balanced by the normal force. Therefore this problem can be equivalently re-expressed as a block brushing past a fixed cylinder in free space.

The essence of this problem is that the block imparts angular momentum to the cylinder, and itself loses an equal amount of angular momentum because what really causes this transfer is the friction, which must be equal and opposite on the two objects by Newton's Third Law

Remember that the velocity of the block has no vertical component at all times as gravity is balanced by the normal force. Therefore this problem can be equivalently re-expressed as a block brushing past a fixed cylinder in free space.

The essence of this problem is that the block imparts angular momentum to the cylinder, and itself loses an equal amount of angular momentum because what really causes this transfer is the friction, which must be equal and opposite on the two objects by Newton's Third Law.