FWIW dad has been fine with like, I'm not sure how to articulate this. But when dd and neighbor got close he asked us if we planned on staying in the neighborhood long term. Because kid had had a lot of upheaval in his life he wanted us to stick around and told us that. And not in a wishful way, almost in a guilt trip way. Dad has no problem letting us watch kid all day or letting kid be enfolded into our family.
This was my thread about this kid asking me to come to his recital from a few months ago:
https://www.dcurbanmom.com/jforum/posts/list/803608.page#15010267
I don't want to undermine dad but it bothers me that he is fine with his kid osmosising into our family and then trying to dictate what we do in our house during the day. I think dd and our family means something to the kid, I don't want to hurt this kid at all.
And to be totally open, kid was visibly upset with nanny when his dad was talking to her, sad and embarrassed. And I don't want myself or nanny to create that embarrassment in him.