This is a question for fun.
Literally, let $\mathcal{F}$ on $X$ be a presheaf which is isomorphic to some sheaf $\mathcal{G}$. Then my question is, among various possible proofs that '$\mathcal{F}$ is also a sheaf', is there a simplest one? What would be? I proved this by categorical method through viewing the definition of 'sheaf' as the exactness of the involved sequence from each open covers $U= \cup_i U_i$. Elementary proof, etc..all possible proofs are welcomed!.
The characterization of sheaves in terms of a certain Lawvere-Tierney topology makes for a fairly short proof. More generally, suppose we have a Lawvere-Tierney topology $j$ on a topos $\mathcal{T}$; then we define a $j$-sheaf to be an object $\mathcal{F}$ of $\mathcal{T}$ such that whenever $V$ is a $j$-dense subobject of $U$, the induced function $\operatorname{Hom}(U, \mathcal{F}) \to \operatorname{Hom}(V, \mathcal{F})$ is a bijection.
Therefore, if we have an isomorphism $\mathcal{F} \simeq \mathcal{G}$ and $\mathcal{G}$ is a $j$-sheaf, then for any such pair $V, U$, we have a commutative square $$\require{AMScd} \begin{CD} \operatorname{Hom}(U, \mathcal{F}) @>>> \operatorname{Hom}(V, \mathcal{F}) \\ @VV \sim V @VV \sim V \\ \operatorname{Hom}(U, \mathcal{G}) @> \sim >> \operatorname{Hom}(V, \mathcal{G}). \end{CD}$$
In this diagram, the bottom, left, and right arrows are bijections; therefore, the top arrow must also be a bijection. This shows that $\mathcal{F}$ is also a $j$-sheaf.
In the specific case of a topological space $X$, the Lawvere-Tierney topology we use is that for $p \in \Omega(U)$, we define $j(p) \in \Omega(U)$ to be the set of $V \subseteq U$ such that there exists an open cover $\{ W_i \mid i \in I \}$ of $V$ with each $W_i \in p$. Thus, if $\mathcal{F}' \subseteq \mathcal{F}$ is a subpresheaf, then the $j$-closure of $\mathcal{F}'$ in $\mathcal{F}$ is the presheaf whose sections over $U$ are all $x \in \mathcal{F}(U)$ such that there exists an open cover $\{ V_i \mid i \in I \}$ of $U$ such that $x |_{V_i} \in \mathcal{F}'(V_i)$ for all $i \in I$. (And then the exactness condition you mention follows from the fact that $\bigcup_{i \in I} h_{V_i}$ is $j$-dense in $h_U$ whenever $\{ V_i \mid i \in I \}$ is an open cover of $U$.)
In fact, it should be straightforward to see how to generalize this to any Grothendieck site.