As it is used here, and by (at least some) topos theorists, a choice object is an object $B$ such that the axiom of choice holds when making choices from $B$.
By contrast, a projective object is an object $A$ such that AC holds when making choices indexed by $A$.
Confusingly, some constructivists use “choice set” to mean what we call a “projective set”.
One way to state the axiom of choice is that every entire relation from $A$ to $B$ (so that every element of $A$ is related to some element of $B$) contains (the graph of) a function $A\to B$. With this terminology, we may say that
Equivalently (at least, in a topos) $B$ is choice iff it has a choice function: a function $c\colon P^+B \to B$ such that $c(x)\in x$ for all $x\in P^+B$. Here $P^+B$ is the object of all inhabited subsets of $B$. We can also say that an object is choice if and only if it is classically well-orderable (that is, it admits a total order where every inhabited subset has a least element); see D4.5.13 in the Elephant.
In constructive mathematics, the principle of excluded middle is equivalent to the statement that the set $2 = \{a \ne b\}$ is a choice set.
Peter Freyd, Choice and Well-ordering , APAL 35 (1987) pp.149-166.
M.-M. Mawanda, Well-ordering and Choice in Toposes , JPAA 50 (1988) pp.171-184.
Peter Johnstone, Sketches of an Elephant vol. II, Oxford UP 2002. (D4.5, pp.987-998)
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