existence of injective function $X\to \Omega^X$

Question

If $\Omega$ is the truth value object from Set and $\Omega^X$ is the exponential of $X$ then I want to show that there exists an injective function from $X$ to its exponential. I've already shown this if I think of $\Omega^X$ concretely as the set of functions from $X$ to $\{t,f\}$. My definition for $\Omega^X$ is based on the evaluation map and exponential (from category theory).... from page 26 of these notes https://www.princeton.edu/~hhalvors/book/sets-new.pdf. I'm not sure how to begin, because I have only a vague understanding of $\Omega^X$ but any hints would be appreciated.

Answer 1

Hint: the usual map we chose is $x\mapsto \{x\}$. This corresponds to the relation $x R f(y)$ iff $x=y$. This yields $X\times X\to \Omega$ (characteristic arrow of a certain subobject) which then yields $X\to \Omega^X$.

Note that if you prove that $\Omega = \{t,f\}$ and that $A^B$ is isomorphic to the set of functions $B\to A$, then the usual proof, the one you probably thought of, works, and you don't need to go back to the definition of exponential object. The advantage of doing it though, is that the proof probably goes through in any topos