Functorial embeddings associated with the Four Subspace Problem

We define a unified categorical framework for studying six subproblems arising from the classical Four Subspace Problem. For each subproblem, we construct a functor from its associated category to the category of representations of the quiver corresponding to the Four Subspace Problem. This approach gives a common structural setting for the six cases considered and allows a simultaneous and coherent analysis via functorial methods. We prove that the six functors are additive and fully faithful, and we show that none of them is dense. As a consequence, each functor induces an equivalence between the corresponding source category and a well-identified full subcategory of the target category. These equivalences provide an effective mechanism for transferring classification results and structural properties, thereby clarifying the structural interrelations among the categories studied.