All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
If by Al-Mishkaat you refer to Mishkaat Al-Masaabeeh, then this book is not one of the books of Ibn Hajar Al-‘Asqalaani, the author of At-Taqreeb. Rather, it was written by Muhammad ibn Abdullah Al-Khateeb Al-‘Umari At-Tabreezi, who died in 741 A.H.
As for the book At-Taqreeb, it was written by Ibn Hajar Al-‘Asqalaani; and it was written after the writing of Al-Mishkaat because Ibn Hajar Al-‘Asqalaani was born in 773 A.H., after the death of At-Tabreezi, as stated by Az-Zirikly in Al-A‘laam. He included At-Taqreeb in the works of Ibn Hajar, but he did not mention a book called Al-Mishkaat among his writings.
It should be noted that Ibn Hajar Al-Haytami Al-Makki explained the book Al-Mishkaat (Sharh Al-Mishkaat), so perhaps the questioner confused him with Ibn Hajar Al-‘Asqalaani.
Sharh Al-Mishkaat was written after At-Taqreeb because Ibn Hajar Al-Haytami came after Ibn Hajar Al-‘Asqalaani, who died in 852 A.H; as stated by Al-Munaawi in Al-Yawaaqeet wa Ad-Durar Sharh Nukhbat Al-Fikar (1/165): “And he (Ibn Hajar Al-‘Asqalaani) died in Thul-Hijjah 852 A.H.”
As regards Al-Haytami Al-Makki, he was born in 909 A.H. and died in 973 A.H.
Mu‘jam Al-Mu’allifeen, written by Ridha Kahhaalah, reads:
“Ahmad ibn Hajar (909 - 973 AH) (1503 - 1566 AD): He is Ahmad ibn Muhammad ibn Muhammad ibn ‘Ali ibn Muhammad ibn ‘Ali ibn Hajar Al-Haytami, As-Sa‘di, Al-Ansaari Ash-Shaafi‘i (Shihaab Ad-Deen, Abul-‘Abbaas) was a scholar of Fiqh involved in different sciences. He was born in the locality of Abu Al-Haytam, in the Gharbiyyah region in Egypt in the month of Rajab, and he died in Makkah.
Among his many works:
- Tuhfat Al-Muhtaaj fee Sharh Al-Minhaaj li An-Nawawi; in the branches of Fiqh of the Shaafi‘i School, in two volumes;
- Mablagh Al-Arab fee Fadhl Al-‘Arab;
- As-Sawaa‘iq Al-Muhriqah li Ikhwaan Al-Ibtidaa‘ wa Adh-Dhalaal wa Az-Zandaqah;
- Ma‘dan Al-Yawaaqeet Al-Multami’ah fee Manaaqib Al-A’immah Al-Arba‘ah;
- Tahreer Al-Maqaal fee Aadaab wa Ahkaam wa Fawaa’id Yahtaaj Ilayha Mu'addibu Al-Atfaal."
It was said that he died in 974, as stated in Al-A‘laam by Az-Zirikli (1/234), and in An-Noor As-Saafir ‘an Akhbaar Al-Qarn Al-‘Aashir (p. 258).”
Allah knows best.