I find it intuitive to define a class of categories that have categorical products and exponentials. However, it is not obvious to me why the additional requirement that they have a terminal object makes the class of categories significantly more interesting.
Why don't people instead place more importance in the class of categories that have products and exponentials but not necessarily terminal objects? What does the terminal object bring us?
Context: I'm very new to category theory.
In a Cartesian closed category, we want a morphism $f:A\to B$ to have some kind of transpose, a morphism with codomain $B^A$. But you only get such a morphism if there is an $X$ with $A\times X\simeq A$, and with a terminal object you get an $X$ that does this with every $A$. With that in place, every $f:A\to B$ corresponds to a unique morphism $\hat{f}:1\to B^A$.
Also intuitively, it seems like if you have a function $f:A\to B$ this should also induce a morphism from $A$ to the set of all constant functions in $B^C$ for any $C$. It would be nice to define constant functions by something like factoring though an object that's like the set of all functions to $B$ from a one-element domain. And we get something like this latter object if we have a terminal object and the exponential $B^1$.