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On pgs. 97-98 of Categories for the Working Mathematician, the author defines the notion of a Cartesian Closed Category:

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The Problem: As far as I recall, the author never defines the notion of an exponential object $c^b$.

Now I'm aware of online definitions of an exponential object -- such as here -- but in this context is it actually required to have a definition of $c^b$? Or is the author just using $c^b$ as notation for the "the image of the functor specified (whatever it may be) by the right adjunction of $- \times b$ (whose counit is similarly notationally referred to as $e : c^b \times b \rightarrow c$)?

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The condition $$\text{hom}(a\times b,c)\cong\text{hom}(a,c^b)$$ (the isomorphism being natural) is the definition of $c^b$. The map $$a\mapsto\text{hom}(a\times b,c)$$ is a contravariant functor from $C$ to $\textbf{Set}$. What Mac Lane is asserting here is that this functor is representable (for all $b$ and $c$) and that we denote a representing object (unique up to isomorphism) by $c^b$.

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