Relationship between power object and exponential object Let $C$ be a category where power objects and exponential objects exist, as well as the terminal object $1$, and the coproduct of $1$ with itself, denoted by $2$.
Let $X$ be an object in $C$. Is there, in general, a relationship between $2^X$ and $P(X)$?
What about if $C$ is a topos?
 A: Just for reference: recall that a power object $P(A)$ comes equipped with a monomorphism $(\in_A)\hookrightarrow A\times P(A)$ such that any monomorphism $r\hookrightarrow A\times B$ for some object $B$ admits a unique $\chi_r:B\to P(A)$ such that
$\require{AMScd}$
\begin{CD}
r @>>> (\in_A) \\
@VVV @VVV \\
A\times B @>>A\times\chi_r> A\times P(A)
\end{CD}
is a pullback square.
On the other hand, the exponential object $2^A$ comes equipped with an evaluation map $\epsilon:A\times2^A\to2$ such that any map $r:A\times B\to2$ factors uniquely through some $\chi:B\to2^A$.
From the perspective of sets, these universal properties say very similar things, but using a different "encoding" of a relation.
For power objects, a relation on $A\times B$ is a subobject ($a$ is related to $b$ if $(a,b)$ lives in the subobject); for exponential objects, a relation on $A\times B$ is a function $A\times B\to2$ ($a$ is related to $b$ if $(a,b)$ gets mapped to $\top\in2=\{\bot,\top\}$).
Let $\mathcal C$ be well-powered, finitely complete, and have all finite coproducts.
Then, we have a functor $\mathrm{Sub}:\mathcal C^{\mathrm{op}}\to\mathbf{Set}$ that sends an object $A$ to the set of subobjects of $A$ (modulo compatible isomorphism), and sends a morphism $f:A\to A'$ to the pullback functor $f^*:\mathrm{Sub}(A')\to\mathrm{Sub}(A)$.
If this functor is representable by some object $\Omega$, then this object is called the subobject classifier.
In other words, this is precisely the object for which monomorphisms into $A$ correspond to maps $A\to\Omega$.
Given a subobject classifier $\Omega$, we can show that a power object $P(A)$ is equivalently an exponential object $\Omega^A$.
Indeed, a monomorphism $(\in_A)\hookrightarrow A\times P(A)$ corresponds to an (evaluation) map $\epsilon:A\times P(A)\to\Omega$, and you can check that the universal properties for each are equivalent.
Now, it's not common for the subobject classifier to be given by the object $2=1+1$ (it is, for instance, in $\mathbf{Set}$).
However, every subobject classifier $\Omega$ comes equipped with canonical maps $\bot,\top:*\to\Omega$ from the terminal object, corresponding to the initial and terminal subobjects of the terminal object.
This induces a reasonable comparison map $2\to\Omega$, which then defines a comparison map $2^A\to\Omega^A\cong P(A)$ for any object $A$.

If you want an explicit example, in a presheaf category $\mathbf{PSh}(\mathcal A)$, the subobject classifier is given---perhaps unsurprisingly---by the presheaf sending $A\in\mathcal A$ to the set of subfunctors of $\mathrm{Hom}(-,A)$.
On the other hand, $2=1+1$ is the constant presheaf on the two-element set, and the map $2\to\Omega$ acts on $A$ by picking out the extreme subfunctors $\varnothing$ and $\mathrm{Hom}(-,A)$ of $\mathrm{Hom}(-,A)$.
