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”.

Properties

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

$A$ is projective iff every entire relation from $A$ to $B$, for any $B$, contains a function $A\to B$, while

$B$ is choice iff every entire relation from $A$ to $B$, for any $A$, contains a function $A\to B$.

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.