Habeas Corpus writs are issued by the courts, not the executive. Any citizen in custody has the right to apply for one in court.
I spoke infelicitously.
There are two documents in question: the court, or the petitioner via the court, requiring the executive to provide evidence justifying the arrest and producing the accused before the court for examination.
It would be better to say the executive "responds" to the petition rather than "provides."
In any case, the way a suspension of heabeas corpus would work in practice is that the executive detains someone, the court is petitioned to require the executive to produce the accused and its evidence, and the executive ignores the summons, pleading that because of public emergency it is released from this obligation.