Given any Heyting algebra $H$, can you construct a category $\mathcal{C}$ such that the the subfunctors of the terminal presheaf on $\mathcal{C}$ (assign singleton to every object) form H?
That is $$\textrm{Sub}_\mathcal{[\mathcal{C}^\textrm{op},\;\mathbf{Set}]}(1)\cong H$$
where $1\colon\mathcal{C}^\textrm{op}\to\mathbf{Set}$ is $1(C)=\{0\}$ for all $C\in\mathcal{C}$.
EDIT: H is a complete Heyting algebra
$\mathsf{Sub}(1)$ is a complete Heyting algebra for any category $\mathcal C$, so there is no choice of $\mathcal C$ that will make $\mathsf{Sub}(1)$ isomorphic to a non-complete Heyting algebra.