Hi I'm having problems with coming up with a proof for this simple property of cartesian closed categories (CCC) and exponential objects, namely that for any object $a$ in a CCC $C$ with an initial object $0$, $a$ is isomorphic to $a^1$ where $1$ is the terminal object of $C$. In most of the category theory books i've read this is usually left as an exercise, but for some reason I can't get a handle on it.
Category Theory – Exponential Objects in Cartesian Closed Category: $a^1 \cong a$
category-theory
Best Answer
For any object $x$, we have: $$\operatorname{Hom}(x,a^1)\cong \operatorname{Hom}(x\times 1,a)\cong \operatorname{Hom}(x,a)$$ So Yoneda's lemma gives us that $a$ is isomorphic to $a^1$.