I am considering the sheaf topos $\mathbf{Sh}(\mathfrak{X})$ on a topological space $\mathfrak{X}$.

Limits in this category are constructed pointwise, as in presheaves. Colimits, however, are not constructed pointwise. As a consequence, the definition of logical operations on the subobject lattices are not as in presheaves, but require closure.

Concretely, given objects $X$ and $Y$ and a subobject $\phi : A \to X \times Y$ the existential $\exists y : Y, \phi(y)$ is constructed as the closure of the image factorization of $\pi_1 \circ \phi$ where $\pi_1 : X \times Y \to X$ is the projection.

I am looking for conditions (on $\phi$ or $X$ and $Y$) to ensure that the image factorization of $\pi_1 \circ \phi$ is as in the underlying presheaves, without explicitly closing. I am happy to assume that $X$ and $Y$ are flabby, if that helps.

For instance, in the sheaf toposes $\mathbf{Sh}(\nu)$ over ordinals $\nu$, equipped with the Alexandrov topology, I think a sufficient (but not necessary) condition is that $\phi$ is $\lnot\lnot$-closed. However this condition is too strong for my needs so I am hoping for something weaker.


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