Since we can likely agree that the O-U or Kaf-K is Kosher
We can agree that something is kosher, yet one of us might want a better hashgocha than something which merely meets minimal standards for “kosher” (I’m not referring to any hashgocha in particular, and my point can be applied to any hashgochas with different standards).
Interestingly, I don’t think there are too many establishments which carry OU and Kof-K, it’s more common to carry a “national” hechsher in addition to a “heimishe” hechsher.