what does it mean for a function "not to respect equality" I am reading from this book about different versions of the axiom of choice.
The author describes weak choice as the following

I have no clue what "respecting equality" means for a function. Surely, if two elements of the domain are actually equal, then they must map to the same element in the range by right uniqueness? Is the author abusing the definition of function to include relations which are not right unique? Or am I not understanding what is meant by 'respect equality'?

I disagree with the interpretation of the author's intent in the other answer. This answer by Andrej Bauer seems more relevant. I'll try to adapt it below, but I highly recommend reading the excellent Bauer post I linked.

Consider sets $X$ and $Y$ each equipped with an equivalence relation, and the axiom:

$\forall R \in |X| \times |Y| . \left[\forall x \in X \, \exists y \in Y . R(|x|,|y|)\right] \implies \left[\exists f \in Y^X \, \forall x \in X . R(|x|,|f(x)|)\right]$

where $|a|$ is the equivalence class of $a$, and $|A|$ is the set of equivalence classes of members of $A$.
If you think of $X$ and $Y$ as sets of "representations" of the "real" objects, and the quotient map to the equivalence classes as mapping representations to the objects, then the axiom states that if $R$ is total, then there exists a choice function $f$ for the representations. There is no condition that $x' \in |x|$ implies that $f(x') \in |f(x)|$.
In the context of a (suitable) type-theory, you indeed get this for free by taking $f$ to be the witness of the assumption $\forall x \in X \, \exists y \in Y . R(|x|,|y|)$.
In contrast, this axiom but with the addition of the condition of respecting equality is too strong: it can be used to decide the halting problem (see Bauer's answer for details).
