Starting in Type-Moon, but as a Index Magician

Morpheus, who had transmigrated into the world of A Certain Magical Index, was still busy cleaning up the mess left by some evil magical cabal when he got caught in a temporal turbulence and was sent flying into the Type-Moon universe. Reborn as an infant, he became the adopted son of the MacRememis family and Bazett's younger brother. Just as he was making preparations for the Fifth Holy Grail War, he discovered that this world was actually the world of FGO, and the Fifth War didn't even exist.

Thus, for the sake of world peace, and to once again pursue the ultimate mystery of magic and walk the path of a Magic God, Morpheus Fraga MacRememis, stepping forth as a Golden-type magician, embarks on the journey to save human history.

The synopsis is weak. Harem, infinite flow. The current world is FGO, while target worlds under consideration include Genshin Impact, Honkai: Star Rail, and series like Demon Slayer.