I came across two different definitions of sheafification and I'm not sure how they are equivalent. One of them is here: About the sheafification
Another one is from Tennison's sheaf theory: Given a presheaf $F$ over $X,$ we construct the sheaf space $LF:=\sqcup_{x\in X}\mathscr{F_{x}}$ which is a disjoint union of stalks and the continuous map is the natural projection $p:LF\rightarrow X $. The open sets are $s[U]=${ $s_{x}\in LF: x\in U$}. Then the sheafification is $\Gamma LF,$ where we construct the sheaf of sections on the sheaf space $LF.$ $\Gamma LF(U)=${$\text{continuous maps} \ \ \sigma :U\rightarrow LF \ \ \text{s.t.} \ \ p(\sigma)=id_{U} $}. So for example, the map $\hat{s}:U\rightarrow LF(U)$ where $x \mapsto s_{x}\in \mathscr{F_{x}}$ would be in $\Gamma LF(U)$. Can someone explain how they are equivalent?