Okay, so these two -- I'm kinda on the fence on how they'd interact (I can actually see it going both ways, tbh, so this idea kinda combines them and I'm not sold on it so we can change it if you think otherwise but okay --)
So, I do think that the five O'Sullivan/Connoghan kids were def raised together (although Annie not so much b/c she's much younger than the rest of them), but everyone except for Sophie was kinda already being ~trained for the time they'd have an active role in the business so yeah, poor Sophie being excluded and like she's the one who most wants this for what it is and not to ~please her father/father figure, but OKAY O'SULLIVAN AND CONNOGHAN KNOW NOTHING. ;D
Anyway, I can see them being friends as kids and Cait kinda passing on what her father had taught her and teaching it to Sophie. At least right up until her father found out what she was doing and was like HELL NO CONNOGHAN DOESN'T WANT YOU FILLING SOPHIE'S HEAD WITH THIS STUFF and so then that would put an end to that and honestly, probs an end to them being close, especially since around that time was probs when Cait stopped having too many friends outside her sister and was just super incredibly focused on her education b/c she knew she wanted to be a lawyer for the family and wanted to get into a good law school and yeah.
So I kinda see them going their separate ways and like, personally Cait thinks there is NO REASON why her father shouldn't have given her a chance b/c HELL YES A WOMAN CAN RUN THIS and I think she feels that she and her sister are proof of this (not that they are gunning for the top spot but just that just b/c Sophie is a girl doesn't mean she can't handle herself).
That being said, she probs does support Jack taking over, if only b/c he does have the experience and everyone would be willing to follow him and that Sophie is ill-prepared (although not for her lack of trying). And she can see that she's trying to prove herself, but in Cait's mind, she's been known to be too hasty (Cait is like SUPER prepared w/ everything and only takes calculated risks/the sure bet) and although, especially with the Sonny w/ her attempt to kill Sonny, it's b/c she hasn't had all the information, she thinks that if ppl know about it w/o all the facts, they may not get behind her and yeah. Cait kinda has complicated feelings about the Sophie thing, haha.
OKAY I THINK THAT'S IT. At the same time, I can also see where they could be friends now or where they could have never really gotten on idk I can't decide PLS HELP.