I feel yields different result than 5 ∪ 7 in the classical set theoretical encoding... I believe 5 ∪ 7 = 7 in the standard encoding of set theory. Because ∪ is the join operation in the natural number lattice (every total order give a lattice structure), yet the lattice structure in ideals of natural number ring is different: the join is LCM and the meet is GCD.

I guess my objection is that the ∪ and ∩ in the set theoretical encoding is rather trivial: the lattice structure in a total order is not terribly informative: join gives the larger element, whereas meet gives the smaller one. Yet the standard encoding of natrual number in category theory (the category generated by one arrow on one object) is slightly more interesting, as composition encodes addition, which is arguably the most interesting opration on natrual numbers.

That being said arguing about encoding of natrual number is not the most informative discussion. but I feel set theory in general is very low level, yet people usually think in more algebraic and high level way, which aligns more closely with category theory.