Categoriсal perspective on the Disjuntion property of the intuitionistic propositional calculus

I came about four different proofs of the disjunction property:

1. formulated in the language of Heyting algebras;
2. done using Kripke models;
3. using the fact that every topological space is an open subspace of a strongly compact space (this can be found in "The Mathematics of Metamathematics" by Helen Rasiowa and Roman Sikorski, IX.6 p.392).

These three appear identical to me (as they all display a minimal non-zero element of an ordered set and use it in a similar way), and it seems like this can be explained by a representation theorem or a Stone Duality.

4. The Gentzen's proof by cut elimination, which doesn't feet this schema but is much easier.

So, my questions are:

1. What is the perspective on the disjunction property of the intuitionistic propositional calculus from the categorical logic point of view? My understanding is that the property is equivalent to the unit filter in the Lindenbaum algebra being prime. This doesn't appear to be leading me anywhere though. Maybe there is a way of proving a similar statement for some bicartesian closed categories?
2. Are there other approaches to proving the disjunction property?

I am mainly concerned with the propositional case with this question, but any similar insight on the disjunction property for predicate logic would be appreciated as well.