# Construction of the equivalence of the finite ordinal category and the category of finite sets

I'm currently reading Mac Lane's "Categories for the Working Mathematician", and on page 18 I found a construction which seems to me to use a universal choice function.

Namely, we work with the categories Finord and Setf, the latter being the category of finite sets and the former the category of sets of the form $$n = \{0, 1, \ldots, n-1\}$$ as a full subcategory of Setf. And in constructing an equivalence of these categories, we choose a function $$\theta_X: X \to \#X$$ for each finite set $$X$$, where $$\# X$$ denotes the cardinality of $$X$$ (which is, of course, an ordinal number in a canonical way).

Here is my question:

Is there any way this may be accomplished using the axiom of choice, or do we need to assume the existence of a global choice function (or a different axiom stronger than the axiom of choice)?

Add using an Easton support product two sets of Cohen subsets to each regular $$\kappa$$, namely force with $$\operatorname{Add}(\kappa,2\times\kappa)$$. In the resulting model there is no uniform way of assigning sets of size $$2$$ a bijection with $$\{0,1\}$$.