Groundwork and observations
Subobjects and quotient objects in arbitrary categories
In any object $X$ in a category $\newcommand{\cat}{\mathsf}\cat{C}$ we can not only consider the partially ordered set $\newcommand{\Sub}{\operatorname{Sub}}\Sub(X)$ of subobjects of $X$, but also the partially ordered set $\newcommand{\Quot}{\operatorname{Quot}}\Quot(X)$ of quotient objects of $X$.
In general, we don’t have a correspondence between the two partially ordered sets $\Sub(X)$ and $\Quot(X)$.
For example, if $X$ is an object of $\cat{Set}$ with $n$ elements, then $\Sub(X)$ has $2^n$ elements, whereas the number of elements of $\Quot(X)$ is given by the $n$-th Bell number.
Subobjects and quotient objects in abelian categories
However, in an abelian category $\cat{A}$ we do have a one-to-one correspondence between $\Sub(X)$ and $\Quot(X)$:
to every subobject $S \to X$ we associate its cokernel $X \to C$, and to every quotient object $X \to Q$ we associate its kernel $K \to X$.
These two assignments are mutually inverse because in an abelian category every monomorphism is the kernel of its cokernel and every epimorphism is the cokernel of its kernel.
We get in this way an anti-isomorphism between $\Sub(X)$ and $\Quot(X)$.
Let us denote this anti-isomorphism by $Φ_X \colon \Sub(X) \to \Quot(X)$.
Subobjects and quotient objects under dualities
Suppose that $D \colon \cat{C} \to \cat{D}$ is a contravariant equivalence of categories (i.e., a duality).
Then $D$ induces isomorphisms (not anti-isomorphisms!) of partially ordered sets
$$
\Sub(X) \to \Quot(D(X)) \,,
\quad
\Quot(X) \to \Sub(D(X)) \,,
$$
which we shall again denote by $D$.
Suppose that $D \colon \cat{A} \to \cat{B}$ is a duality between abelian categories.
Then $D$ interchanges kernels and cokernels.
We thus find that the following diagram commutes:
\begin{equation}
\require{AMScd}
\begin{CD}
\Quot(X) @>{D}>> \Sub(D(X)) \\
@V{Φ^{-1}_X}VV @VV{Φ_{D(X)}}V \\
\Sub(X) @>>{D}> \Quot(D(X))
\end{CD}
\tag{$\ast$}
\end{equation}
Radicals and socles from an order-theoretic perspective
Given a partially ordered set $P$ let us make the following auxiliary definitions:
An element of $P$ is proper if it is not the greatest element of $P$.
An element of $P$ is maximal if it is maximal among all proper elements of $P$.
The radical of $P$ is the infimum of all proper elements of $P$, and it is denoted by $\newcommand{\Rad}{\operatorname{Rad}} \Rad(P)$.
An element of $P$ is non-trivial if it is not the least element of $P$.
An element of $P$ is minimal if it is minimal among all non-trivial elements of $P$.
The socle of $P$ is the supremum of all minimal elements of $P$, and it is denoted by $\newcommand{\Soc}{\operatorname{Soc}} \Soc(X)$.
With these auxiliary definitions, we have in any abelian category $\cat{A}$ the equalities of subobjects of $X$,
$$
\Rad(X) = \Rad(\Sub(X)) \,,
\quad
\Soc(X) = \Soc(\Sub(X)) \,.
$$
Radials and socles under (anti-)isomorphisms
An isomorphism of partially ordered sets preserves radicals and socles.
An anti-isomorphism, on the other hand,
- interchanges the greatest element with the least element;
- consequently interchanges proper elements with non-trivial elements;
- consequently interchanges maximal elements with minimal elements;
- interchanges infima with suprema;
- consequently, interchanges the socle with the radical.
Simple objects
An object $X$ of a category $\cat{C}$ is called simple if $\Quot(X)$ consists of precisely two objects.
In an abelian category, this is equivalent to saying that $\Sub(X)$ has precisely two objects (because of the anti-isomorphism between $\Sub(X)$ and $\Quot(X)$).
Consequences
Let $D \colon \cat{A} \to \cat{B}$ be a duality between abelian categories,
I am trying to prove that the following two objects are isomorphic for any object $\newcommand{\Hd}{\operatorname{Hd}}X ∈ \cat{C}$:
$$
D(\Soc(X)) ≅ \Hd(D(X)) \,.
$$
Let us now consider the commutative diagram $(\ast)$ of partially ordered sets.
The horizontal arrows are isomorphisms, whereas the vertical arrows are anti-isomorphisms.
We have therefore the following equalities of quotient objects of $D(X)$:
\begin{align*}
\Hd(D(X))
&=
D(X) / \Rad(D(X)) \\
&=
Φ_{D(X)}( \Rad(D(X)) ) \\
&=
Φ_{D(X)}( \Rad(\Sub(D(X))) ) \\
&=
Φ_{D(X)}( D(\Rad(\Quot(X))) ) \\
&=
D( Φ^{-1}_X( \Rad(\Quot(X)) ) ) \\
&=
D( \Soc(\Sub(X)) )\\
&=
D( \Soc(X) ) \,.
\end{align*}
How do I show from this that for any simple object $S ∈ \cat{C}$ we have $D(S) ≅ S$?
An object $S$ of $\cat{A}$ is simple in $\cat{A}$ if and only if $D(S)$ is simple in $\cat{B}$, because of the isomorphism $\Sub(D(X)) ≅ \Quot(D)$.
So if $S$ is simple, then $D(S)$ is again simple.
But I don’t think that we can show $S ≅ D(S)$ for $D \colon \cat{A} \to \cat{A}$, even if $ξ \colon D^2 ≅ \mathrm{Id}_{\cat{A}}$ with $\mathrm{id}_{D(X)} = D(ξ_X) ∘ ξ_{D(X)}$.
To see this, consider the group $G = ℤ/n$ for $n ≥ 3$ and let $\cat{A}$ be the category of finite-dimensional complex representations of $G$.
The category $\cat{A}$ is essentially small and $ℂ$-linear with finite-dimensional $\mathrm{Hom}$-spaces.
(And according to Maschke’s theorem, $\cat{A}$ is semisimple.)
The category $\cat{A}$ admits the auto-equivalence $D ≔ (-)^*$ with $D^2 ≅ \mathrm{Id}_{\cat{A}}$.
More explicitly, we have the natural isomorphisms
$$
ξ_V \colon V \longrightarrow V^{**}
$$
given by
$$
ξ_V(v)(v^*) = v^*(v)
$$
for all $v ∈ V$, $v^* ∈ V^*$.
These isomorphisms satisfy the equation $\mathrm{id}_{V^*} = (ξ_V)^* ∘ ξ_{V^*}$ because
$$
(ξ_V)^*\Bigl( ξ_{V^*}( v^* ) \Bigr)(v)
=
\Bigl( ξ_{V^*}( v^* ) ∘ ξ_V \Bigr)(v)
=
ξ_{V^*}( v^* )( ξ_V(v) )
=
ξ_V(v)(v^*)
=
v^*(v)
$$
for all $v ∈ V$, $v^* ∈ V^*$.
We have for every $n$-th root of unity $λ$ a one-dimensional representation $V_λ$ of $G$, on which the cyclic generator $[1]$ of $G$ acts by multiplication with $λ$.
These representations are pairwise non-isomorphic, every irreducible complex representations of $G$ is of this form, and we have $(V_λ)^* ≅ V_{\overline{λ}}$ for every $n$-th root of unity $λ$.
Therefore, $D(V_λ) ≅ V_λ$ if and only if $λ = \overline{λ}$, which happens only for $λ = 1$ (the trivial representation) and $λ = -1$ (the alternating representation, existing only if $n$ is even.)
So, since $n ≥ 2$, we have simple objects $S$ in $\cat{A}$ with $D(S) ≇ S$.
