Do we have $\mathcal G=\mathcal H$?

Question

Let $X$ be a scheme and $\mathcal F$ an $\mathcal O_X$-module. We let $\mathcal G=(\mathcal F\otimes_{\mathcal O_X}\mathcal F)\otimes_{\mathcal O_X}\mathcal F$, and let $\mathcal H$ be the sheaf associated to the presheaf $U\mapsto \mathcal (F(U)\otimes_{\mathcal O_X(U)}\mathcal F(U))\otimes_{\mathcal O_X(U)}\mathcal F(U)$, where $U$ is any open set of $X$, do we have $\mathcal G=\mathcal H$?