Value of short call option

Dear tutor,
Could you, please, help me with the following problem.

In BPP study text they say that the value of a short call option increses when the value of the underlying decreases. I cannot understand why this is so.

Let`s consider two examples. We sold a European call options (i.e. we are short) with the exercise price = $5. This means that we took the obligation to buy the underlying at $5.
Suppose that at the expiration date the value of the underlying is
a) $4
b) $6

The ouycome for us wll be:
a) as the value of the underlying is less than the exercise price the buyer of our call option will exercise it as it is more profitable for him to sell the underlying to us at $5 than to sell it on the market at $4. Therefore we incure loss.
b) as the value of the underlying is more than the exercise price the buyer of our call option will not exercise it as it is more profitable for him to sell the underlying to the market at $6 than to sell it to us at $5.
Thus my understanding is contrary to that of BPP. Where am I wrong?