If $j$ is a Grothendieck-Lawvere-Tierney topology on an elementary topos ${\bf E}$, it is by now well known that the inclusion $\text{sh}_j({\bf E})\rightarrowtail{\bf E}$ of the full subcategory of $j$-sheaves has a left exact left adjoint ${\bf E}\rightarrow\text{sh}_j({\bf E})$, called the associated sheaf functor. Two sorts of construction for the associated sheaf functor are known: a two-step construction, by which one first half-sheafifies an object $X$, obtaining a $j$-separated object $X^+$ that is already a sheaf if $X$ was separated, and then half-sheafifies again, verifying that the sheaf $(X^+)^+$ is the desired associated sheaf; and a one-step construction, described elsewhere by the authors. Here the authors give a direct internal description of $X^+$ as a subobject $X^+\rightarrowtail\Omega^X$ of the power object $\Omega^X$ of $X$: Where $\tilde X\rightarrowtail\Omega^X$ is the object classifying partial maps to $X$, and where $J_X\rightarrowtail\Omega^X$ is the subobject classified by $\Omega^X\underset\exists_{X\rightarrow 1}\to\rightarrow\Omega\underset j\to\rightarrow\Omega$, the authors define $X^+$ as the image of the composition $\tilde X\cap J_X\rightarrow\Omega^X\overset j^X\to\rightarrow\Omega^X$ (which is the same as the intersection $J_X\cap I_X$ of $J_X$ with the image $I_X$ of the composition $\tilde X\rightarrowtail\Omega^X\underset{j^X}\to{\rightarrowtail}\Omega^X$). For the sake of comparison, P. T. Johnstone's now standard reference [{\it Topos theory}, especially pp. 84 ff., Academic Press, London, 1977; MR0470019 (57 \#9791)] takes $X^+$ as the direct limit of an internal presheaf $\hat X$ defined on the internal poset $J_1$. \par The authors view their description of $X^+$ as being more explicit or transparent and permitting the geometric intuition to lead more directly to straightforward proofs of the lemmas one needs along the way to the proof that $(X^+)^+$ is the associated sheaf. In any event, it is precisely with these arguments, in the framework of this description of $X^+$, that the bulk of the present work is concerned.