- Show that every slice category $\mathbb{Sets}/X$ is cartesian closed. Calculate the exponential of two objects $A \to X$ and $B \to X$ by first determining the Yoneda embedding $y : X \to \mathbb{Sets}/X$ , and then applying the formula for exponentials of presheaves. Finally, observe that $\mathbb{Sets}/X$ is a topos, and determine its subobject classifier.
I feel like using the equivalence relation between $\mathbb{Sets}/X$ and $\mathbb{Sets}^{X}$, but I'm curious about the hint given, regarding $y: X \to \mathbb{Sets}/X$, since the set $X$ contains no non-identity arrows when regarded as a category. How is this related to the problem?
Thanks!