Geraldine Brady in pages 132–133 of her book "From Peirce to Skolem: A Neglected Chapter in the History of Logic" said that the first proposal for an axiomatic set theory was actually by Charles Sanders Peirce in his article "On the Algebra of Logic: A Contribution to the Philosophy of Notation".

There is a second issue which is independent of the identity of objects. When the objects of a category no longer form a set, something else needs to replace the set, and most proposals these days say the objects of a category should form a type or $\infty$-groupoid. The problem is that the resulting naive definition of category results in a "precategory", a Segal space whose hom-spaces are sets. To actually get a category one actually needs an additional Segal condition on a precategory to get a category. But that Segal condition isn't true in dagger categories.

as defined above the dagger operations in a dagger category are a family of functions between hom-sets, rather than a contravariant functor. The resulting daggers are similar to composition of morphisms in a category in that regards. As a result, there is no strict identity on objects, only isomorphisms and unitary isomorphisms between objects.

the natural numbers $(N, \mathrm{pt}, \mathrm{succ})$ as defined by the second-order Peano axioms or by a universal property in the category of sets could refer to either. It is really the semigroup structure resulting from the recursive/inductive definition of addition that determines whether $\mathrm{pt}$ is zero or one.