  The notion of an internal preneighbourhood space on a finitely complete
category with finite coproducts and a proper $(\mathsf{E}, \mathsf{M})$ system
such that for each object $X$ the set of $\mathsf{M}$-subobjects of $X$ is a
complete lattice was initiated in \cite{2020}. The notion of a closure
operator, closed morphism and its near allies investigated in \cite{2021-clos}.
The present paper provides structural conditions on the triplet $(\mathbb{A},
\mathsf{E}, \mathsf{M})$ (with $\mathbb{A}$ lextensive) equivalent to the set
of $\mathsf{M}$-subobjects of an object closed under finite sums. Equivalent
conditions for the set of closed embeddings (closed morphisms) closed under
finite sums is also provided. In case when lattices of admissible subobjects
(respectively, closed embeddings) are closed under finite sums, the join
semilattice of admissible subobjects (respectively, closed embeddings) of a
finite sum is shown to be a biproduct of the component join semilattices.
Finally, it is shown whenever the set of closed morphisms is closed under
finite sums, the set of proper (respectively, separated) morphisms are also
closed under finite sums. This leads to equivalent conditions for the full
subcategory of compact (respectively, Hausdorff) preneighbourhood spaces to be
closed under finite sums.
