Did the first man give woman her self-understanding?

Ortlund has written chapter three of Recovering Biblical Manhood and Womanhood CBMW’s book. In this chapter, Ortlund says:
In designating her “Woman” the man interprets her identity in relation to himself. Out of his own intuitive comprehension of who she is, he interprets her as feminine, unlike himself, and yet as his counterpart and equal. Indeed, he sees in her his very own flesh. And he interprets the woman not only for his own understanding of her, but also for her self-understanding. God did not explain to the woman who she was in relation to the man, although He could have done so. He allowed Adam to define the woman, in keeping with Adam’s headship. (emphasis mine)
Let me ask a couple of questions. Who explained to the man who he himself was for his own self-understanding? …