Let $H$ be a complete Heyting algebra, i.e, a complete poset which is also cartesian closed as a category. In particular, this algebra has a least and a greatest element. We can define the category of $H$-sets as the category consisting of pairs $(X,\xi)$, where $X$ is a set and $\xi$ a map from $X$ to $H$. A morphism of $H$-sets $(X,\xi)$ and $(Y,\upsilon)$ is a map $f:X\to Y$ such that for every element $x\in X$ we have $\xi(x)\leq \upsilon(f(x))$. It can be shown that the category $H\mathbf{Sets}$ is complete rather easily. Also the proof of cocompleteness does not take a lot of effort.
However, I was wondering if there exists a way to proof the cocompleteness of $H\mathbf{Sets}$ by just using the fact that for every arbitrary Heyting algebra $H$ the category $H\mathbf{Sets}$ is complete. That is, can we in some way construct a Heyting algebra $H'$ such that $H\mathbf{Sets^{op}}$ is isomorphic with $H'\mathbf{Sets}$ as categories? Curious to hear some thoughts about this.
Thanks in advance!
Your suggestion is not possible, since $H$-Set has a strict initial object: the initial object $(\emptyset,\emptyset)$ admits no maps from any other object. However the terminal object $(*,*\mapsto \top)$ admits many maps to other objects, so in the opposite category the initial object is not strict. Generally, it does not occur that classes of categories similar to Set are closed under taking opposites. This is largely a phenomenon reserved for the world of homological algebra, though even there many natural classes are not so closed.