Exponential objects in a cartesian closed category: $a^1 \cong a$ 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.  

I know that this is a bit late, but I'm readong the book right now and found this post, so i'll give my solution anyway, which doesn't use the Yoneda Lemma, or adjunctions, etc. 
The idea is to notice that $id_a : a\to a$ is a terminal object in the comma category $C\to a$. But the definition of the exponential allows you to show that $ev : a^1 × 1 \to a$ is also a terminal object, and therefore isomorphic to the first one. It is then easy to conclude that $a^1×1 \simeq a$ and so $a^1 \simeq a$
