Closure of the product of subfunctors

Background:

• Let $X: textbf{CRing} to textbf{Set}$ be a presheaf on the category of affine schemes and $Z subseteq X$ a subfunctor. One defines $Z$ to be closed if for every ring $A$ and every morphism $f: text{Hom}(A , -) to X$ the inverse image $f^{-1}(Z)$ is of the form $R mapsto { varphi : A to R | varphi(I) = 0 }$ for some ideal $I subseteq A$.
• The intersection of subfunctors is defined naively, as is the closure (denoted by $overline{Z}$) of a subfunctor $Z subseteq X$ (it is the intersection of all closed subfunctors of $X$ containing $Z$).
• If $Y$ is another presheaf, the product of $X$ and $Y$ is also defined naively.

Context: In section 1.14 of Jens Jantzen’s great book "Representations of Algebraic Groups", the following is stated: If $X$ and $Y$ are presheaves which are schemes over a noetherian ring $k$ and $Z subseteq X$ is a subscheme, and if $Z, X$ are algebraic and $Y$ is flat, then $overline{Z times Y} = overline{Z} times Y$. For the proof, he references Demazure-Gabriel I, section 2, 4.14 (although in my copy of Bell’s translation this reference unfortunately doesn’t exist).

Actual Question:
Is this true for general presheaves? I.e. if $X$ and $Y$ are presheaves and $Z subseteq X$ is a subfunctor, is it true that $overline{Z times Y} = overline{Z} times Y$? I worry that it isn’t because of the conditions in Jantzen stated above, but I haven’t been able to decide either way. (Also side question: does anyone know the correct reference in the translation?)