I need to show that the existence of a sheafification for every presheaf $\mathcal{O}'$. I know that it's enough to show that the covariant functor $$Hom(\mathcal{O}' , ·) :\textbf{Sh}_X\longrightarrow \textbf{Set}$$ is representable in $\textbf{Sh}_X$, where $X$ is some topological space,and $\textbf{Sh}_X$ denotes the category of sheaves of rings on $X$ I tried to use the following :
For every open set $U$ in $X$ let $\mathcal{O}(U)$ be the set of maps $$g : U\longrightarrow \bigcup_{x\in X}\mathcal{O}_{x} ^{'} $$, such that for every $x \in U$, we have $$g(x) \in \mathcal{O}_{x} ^{'} $$ and there exists a neighborhood $V$ of $x$ contained in $U$ and a section $ s \in \mathcal {O}'(V) $ such that for all $ y \in V$, $g(y) = [s]_y$. (Here, $\mathcal{O} _{X} ^{'}$ is the stalk of $\mathcal{O}'$ at $x$ and $[s]_y$ is the germ of $s$ at $y$). I need some body to explain how to complete the proof using this.