Functorial isomorphisms $\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\PSh{PSh}$We know that a presheaf $\mathcal{F}$ on category $ \mathcal{C} $  gives a colimit preasheaf $ \mathcal{F}^{+} $ defined by Čech cohomology, and then we have the following theorem:
$ \mathcal{F}^{\sharp} = (\mathcal{F}^{+})^{+} $ is a sheaf and the canonical map induces a functorial isomorphism $ \Hom_{\PSh(\mathcal{C})}(\mathcal{F} , \mathcal{G}) = \Hom_{\Sh(\mathcal{C})}(\mathcal{F}^{\sharp} , \mathcal{G}) $,
for all $ \mathcal{G} $ in $ \Sh(\mathcal{C}) $.
My question is, can we can find a functorial isomorphism between
$ \Hom_{\PSh(\mathcal{C})}(\mathcal{F}^{+} , \mathcal{G}) $ and  $ \Hom_{\Sh(\mathcal{C})}(\mathcal{F}^{\sharp} ,  \mathcal{G}) $?
And the analogous question for $ \Hom_{\PSh(\mathcal{C})}(\mathcal{F} , \mathcal{G}) $ and $ \Hom_{\Sh(\mathcal{C})}((\mathcal{F}^{\sharp})^{+} ,  \mathcal{G}) $.
 A: Since Čech cohomology is mentioned, I presume that $C$ is the category of open subsets of a topological space.
More generally, we can assume $C$ to be an arbitrary site.
In this case, the answer to both questions is yes.
It follows from a more general result:
If $G$ is a sheaf and $f\colon P\to Q$ is a local isomorphism of presheaves
(i.e., a morphism that becomes an isomorphism after passing to associated sheaves; in the case of sites with enough points, such as topological spaces, it can be described as a morphism that induces an isomorphism on all stalks),
then the induced map
$$\def\Hom{\mathop{\rm Hom}} \Hom(Q,G)→\Hom(P,G)$$
is an isomorphism.
This abstract result applies to the two cases under consideration
because the maps $F^+→F^\sharp$ and $F\to (F^\sharp)^+=F^\sharp$
are local isomorphisms.
Indeed, even more generally, the natural map $F\to F^+$ is a local isomorphism for any presheaf $F$.
Indeed, the associated sheaf functor can be computed as $F↦F^{++}$,
so if we apply it to the morphism $F→F^+$, we get the identity map $F^{++}→F^{+++}=F^{++}$.
In the case of sites with enough points, this also follows immediately from the fact that the stalk functor is cocontinuous, in particular, it preserves the colimit used to define $F^+$.
