Exponential objects and hom-sets.

Question

Let $C$ be a cartesian closed category and $X, Y$ two objects of $C$.

Is it the case that $\text{Hom}(X,Y) = Y^X$?

Answer 1

No, since $\hom(X,Y)$ is a set, but $Y^X$ is an object of $C$.

The relation between these two guys is given by the following formula (where $1$ denotes a terminal object): $$\hom(1,Y^X) \cong \hom(X,Y).$$ This bijection follows immediately from the definitions.

In other words, $Y^X$ is an object whose set of "global sections" is $\hom(X,Y)$. This is true verbatim if $C$ is a Grothendieck topos.