Shape of the Yield Curve

I am not sure that I comprehended the below sentence correctly;

"As an upward-sloping yield curve becomes flatter, the call option value increases."

Did the author mean flatter or downward sloping?

Flatter, as far as I understand means that the interest rates are similar along the yield curve where the current interest rates are close to the one which calculated the par value. Accordingly, the call option will not be excerised.

However, if the yield curve is downward sloping, the interest rates will decrease and accordingly the bond value will be higher than the par value and the call option will be in the money.

Do I correctly understand it?

Your explanation is too complex. I think what you are explaining is a flat yield curve, not flatter. When something becomes flatter, it is a movement.

Intially the Curve is upward sloping. As it becomes flatter, the call option increases.

It doesn’t have to be downward sloping.