How to prove this Category Theory theorem? 
I don't know from where to approach this theorem. Maybe from that relationship between representable function and left adjoints, but I still can't properly relate to it.
 A: Disclaimer: the following is not a real answer but more an addendum to address a comment in the said answer.
As told before the claim in the question is false.
What is true instead is that: 
If $H \colon \mathcal D \to \mathcal C$ is a functor then the following are equivalent


*

*for all $B \in O_\mathcal{C}$ the functor $MOR(B,H-)$ is representable

*there is a function $F \colon O_\mathcal{C} \to O_\mathcal{D}$ such that $MOR(B,H-)\cong MOR(F(B),-)$ for every $B \in O_\mathcal{C}$

*there is a functor $F \colon \mathcal C \to \mathcal D$ such that there is a natural isomorphism $MOR(-,H-)\cong MOR(F-,-)$ between these functors from $\mathcal C^d \times \mathcal D$ into $\mathbf{Set}$ (by the way I assume that $\mathcal C^d$ means the opposite of the category $\mathcal C$).


The equivalence of 1 and 2 is trivial: it follows from the general logical equivalence 
$$\forall x \in A\ \exists y \in B P(x,y) \iff \exists f \colon A \to B \ \forall x \in A P(x,f(x))$$
In details 1. can be read as $\forall B \in O_\mathcal{C}\ \exists a Y \in \mathcal{D} MOR(B,H-)\cong MOR(Y,-)$ while 2. is $\exists F \colon O_\mathcal{C} \to O_\mathcal{D}\ \forall B \in O_\mathcal{C} MOR(B,H-)\cong MOR(F(B),-)$.
Equivalence between 2 and 3 is more interesting as it can be seen as an application of Yoneda. 
As stated in the previous answer 3 trivially implies 2: indeed if $F$ is such that $MOR(-,H-)\cong MOR(F-,-)$ the for each $B \in \mathcal C$ we have $MOR(B,H-)\cong(F(B),-)$, so the object function underlying $F$ is the one that makes 2 true.
For the other implication let $F \colon \mathcal{C} \to \mathcal{D}$ be a function such that $(\varphi_B \colon MOR(B,H-)\cong MOR(F(B),-))_{B \in \mathcal{C}}$ is a family of natural isomorphisms between functors from $\mathcal D$ to $\mathbf{Set}$.
We would like to extend $F$ to a functor, in order to do that we can observe that for every morphism $f \colon B \to B'$ in $\mathcal C$
we have the following diagram of $\mathbf{Set}$-valued functors.
$$
\require{AMScd}
\begin{CD}
MOR(F(B'),-) @>\varphi_B'^{-1}>> MOR(B',H-) \\
@V?VV                            @VV{MOR(f,H-)}V \\
MOR(F(B),-)  @<<\varphi_B<      MOR(B,H-)  
\end{CD}
$$
Now it is a known fact (and a consequence of the Yoneda Lemma) that the functor sending every $D \in O_\mathcal{D}$ into $MOR(D,-)$ is fully-faithful embedding, this implies that every natural transformation of the kind $MOR(D',-) \to MOR(D,-)$ is of the form $MOR(f,-)$ for some $f \colon D \to D'$.
In the case at the hand the previous remark tells us that the natural isomorphism $?$ in the previous diagram is of the form $MOR(F(f),-)$ for some $F(f) \colon F(B) \to F(B')$ in $\mathcal D$.
This gives us an extension of the function $F$ between the sets $O_\mathcal{C}$ and $\mathcal D$ to morphisms between the graphs of $\mathcal C$ and $\mathcal D$.
Actually these data gives a functor: for every pair $f \colon B \to B'$ and $g \colon B' \to B''$
$$MOR(F(g\circ f),H-)=\varphi_{B''}\circ MOR(g \circ f,H-)\circ\varphi_B^{-1}$$
$$=\varphi_{B''}\circ MOR(f,H-)\circ \varphi_{B'}^{-1}\circ\varphi_{B'}\circ MOR(g,H-)\circ \varphi_B$$ 
$$=MOR(F(f),-)\circ MOR(F(g),-)=MOR(F(g)\circ F(f),-)$$
and so $F(g\circ f)=F(g)\circ F(f)$.
Proving that $F(\text{id}_B)=\text{id}_{F(B)}$ is a similar calculation.
Finally you can observe that the mappings $(\varphi_{B,D} \colon MOR(B,HD)\cong MOR(FB,D))_{B \in \mathcal{C},D \in  \mathcal{D}}$ are natural in $D$ by hypothesis (the $\varphi_B$ are natural transformation between $MOR(B,H-)$ and $MOR(FB,-)$)  but also in $B$ (because of how we defined $F(f)$: as the only morphism that made commute the diagram above which express naturallity in $B$).
So $\varphi \colon MOR(-,H-)\cong MOR(F-,-)$ give the wished natural isomorphism.
A: Actually the claim, as it is stated, is false: it tells that those sentences should be equivalent for every pair of functors.
The problem is the following: the claim of your statement says

Given two functors $H \colon \mathcal D \to \mathcal C$ and $F \colon \mathcal C \to \mathcal D$, whatever they are, if these two functors satisfy one of the following conditions,$(i)$ $MOR(B,H-)$ is representable for every $B \in O_{\mathcal C}$,$(ii)$ $MOR(B,H-)\cong MOR(F(B),-)$ for every $B \in O_{\mathcal C}$ and $(iii)$ there is a natural isomorphism $MOR(-,H-)\cong MOR(F-,-)$, then the other two hold too.

It is true that $(iii)$ implies $(ii)$ and that $(ii)$ implies $(i)$, the inverses do not hold. 
If the claims above where true then for instance we should have that if $F \colon \mathcal C \to \mathcal D$ is any functor and $H \colon \mathcal D \to \mathcal C$ is a functor such that $MOR(B,H-)$ is representable then $MOR(B,H-)$ should be naturally isomorphic to $MOR(F(B),-)$ (this would be an application of $(i) \Rightarrow (ii)$). 
There is no reason that should be true if $F$ is any functor. Of course it is true that if $MOR(B,H-)$ is representable for every $B \in  O_\mathcal{C}$(that is $(i)$) then there is a functor $F \colon \mathcal C \to \mathcal D$ such that $(ii)$ and $(iii)$ are true: it sufficies to take $F$ to ba a left adjoint to $H$.
In the very same spirit if it was true that $(ii)$ implies $(iii)$ then  every functor $F \colon \mathcal C \to \mathcal D$, such that $MOR(B,H-)$ is naturally isomorphic to $MOR(F(B),-)$, should be a left adjoint to $H$ (this is $(iii)$) which is not true in general. 
Building counter-examples is not really that hard harder than I thought.
By the way the other implications are trivial:


*

*that $(iii)$ implies $(ii)$ follows from the fact that if $\varphi \colon MOR(-,H-) \cong MOR(F-,-)$ is a natural isomorphisms between functors from $\mathcal C^\text{op} \times \mathcal D$ to $\mathbf{Set}$ then specializing $\varphi$ at a $B \in O_{\mathcal C}$ you get the natural isomorphism $\varphi_{B,-} \colon MOR(B,H-)\cong MOR(F(B),-)$;

*$(ii)$ implies $(i)$ because if $\varphi_B \colon MOR(B,H-)\cong MOR(F(B),-)$ is natural isomorphism then, by definition, $MOR(B,H-)$ is representable (represented by $F(B)$).


Edit: since the OP asked in what follows I'll give some counter-examples to the wrong implications.
$[(i) \not \Rightarrow (ii)]$: Let consider $\mathcal C=\mathcal D=\mathbf {Set}$, $H=\text{id}_\mathbf{Set}$, and $B$ a non empty set so $MOR(B,H-)=MOR(B,-)$, and let $F \colon \mathbf{Set} \to \mathbf{Set}$ be the constant functor defined by the equations
$$F(X) = \emptyset$$
$$F(f) = \text{id}_\emptyset$$
for every $X \in \mathbf{Set}$ and $f \colon X \to Y$, for opportune sets $X$ and $Y$.
Clearly $MOR(B,-)$ is representable (it is represented trivially by $B$ it self), but there can be no natural isomorphism between $MOR(B,-)$ and $MOR(F(B),-)=MOR(\emptyset,-)$ because by yoneda if a such isomorphism exists the objects $B$ and $\emptyset$ should be isomorphic, which is not possible since $B$ is not empty by hypothesis.
Proving that $[(iii) \not \Rightarrow (ii)]$ is a little bit harder... possibly it will require a little more time to find a counterexample so be patient. :)
