From Mac Lane's Category Theory:
What does the symbol $c^b$ mean in the red circle in the picture below?
I see that for $\langle F, G, \phi \rangle: C \rightarrow C$ to be an adjoint it must be the case that $\phi_{x,a}=C(x \times b, a) \cong C(x, a^b)$
but if $b$ and $a$ are objects, what is meant by an object $a^b$?
To spell out Arnaud D.'s comment...
It's just a name. The exact same thing is happening with $a\times b$ and with $t$.
As Arnaud D. states, we are assuming the existence of a right adjoint to (among other things) $-\times b$. That is, we're assuming that for each $b$ there is a functor $F_b$ such that $-\times b \dashv F_b$. We simply choose to write $F_b(a)$ as $a^b$ as that's more evocative.