The following theorem is from Peter Smith's Gentle Intro to Category Theory (P119):
If there exists an exponential of $C$ by $B$ in the category $C$ , then, for any object $A$ in the category, there is a one-one correlation between arrows $A \times B \to C$ and arrows $A \to C^B$.
There is also a one-one correlation between arrows $A\to C^B$ and arrows $B\to C^A$.
Proof: By definition of the exponential $[C^B, ev]$, an arrow $g: A\times B \to C$ is associated with a unique 'transpose' $\bar{g}:|A \to C^B$ making the diagram commute.
The map $g\mapsto \bar{g}$ is injective. For suppose $\bar{g}=\bar{h}$. Then $g=ev \circ (\bar{g}\times 1_B)=ev\circ (\bar{h}\times 1_B)=h$.
The map $g\mapsto \bar{g}$ is also surjective. Take any $k: A\to C^B$, then if we put $g=ev\circ (k \times 1_B)$,$\bar{g}$ is the unique map such that $ev\circ (\bar{g}\times 1_B)=g$, so $k=\bar{g}$.
Hence $g\mapsto \bar{g}$ is the required bijection between arrows $A\times B\to C$ and arrows $A\to C^B$, giving us the first part of the theorem.
For the second part, we just note that arrows $A\times B \to C$ are in one-one correspondence with arrows $B\times A \to C$, in virtue of the isomorphism between $A\times B$ and $B\times A$. We then apply the first part of the theorem.
I am having trouble understanding this proof, because I don't understand what 'one-one correlation/correspondence' is. I searched throughout the book but this theorem (and another which uses this theorem) is the only place where this notion is mentioned.
It does not seem like Smith is referring to one-one in terms of injectivity either, because he explicitly uses it to describe $g\to \bar{g}$. So what is he referring to here (and what is he trying to do)?