How unique is the exponential of sets?

Question

Firstly I would like to thank everyone in this site for their valuable help. You have helped me a lot understanding the idea of an exponential.

I would like to ask a final question that is the last piece of the puzzle for me. It is clear that the function set $\text{Hom}(X,Y)$ is the exponential (by which I mean the categorical exponential, as defined by the universal property of the evaluation map) of the sets $X$ and $Y$ in the category of sets. What does the word "is" mean in the previous sentence? What kind of uniqueness exists here? Are all sets that are isomorphic to $\text{Hom}(X,Y)$ also exponentials of $X$ and $Y$?

Answer 1

Don't forget that the exponential consists of an object $Y^X$ and an evaluation map $e : Y^X \times X \to Y$.

Every isomorphism $i : Y^X \to Z$ lets us construct another exponential whose object is $Z$ and whose evaluation map is $e \circ (i^{-1} \times 1_X)$. Every other exponential is of this form.

In fact, one can give a suitable definition of a morphism of exponentials, and all exponentials are isomorphic.

(technically, we should be allowing for the choice of product to vary too)

Answer 2

For a set $S$, the exponentiaton $(-)^S$ is right adjoint to $- \times S$.

It is a direct consequence of Yoneda's lemma that adjoints are unique up to isomorphism. See this answer.