Additive cocontinuous module functors and their schematic category

Let $A$ and $B$ be unital rings, and let $A\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ and $B\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ be their categories of unital left modules (Unital left and right modules over a ring; unqualified module means left module). A functor $F:A\text{-}\mathbf{M}\mathbf{o}\mathbf{d}\to B\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ is additive when its induced maps on hom-groups are group homomorphisms (Additive functor), and cocontinuous when it preserves every small (set-indexed) colimit (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors). The functor is additive cocontinuous when it is both additive and cocontinuous.

Cocontinuity has a module-theoretic reformulation: an additive functor $F:A\text{-}\mathbf{M}\mathbf{o}\mathbf{d}\to B\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ preserves all small colimits if and only if it is right exact (Left exact and right exact functors) and preserves arbitrary direct sums. The characterization is proved in An additive module functor is cocontinuous exactly when it is right exact and preserves coproducts ↗.

We write ${\mathrm{F}\mathrm{u}\mathrm{n}}_{\mathrm{a}\mathrm{d}\mathrm{d}}^{\mathrm{c}\mathrm{o}\mathrm{c}}(A\text{-}\mathbf{M}\mathbf{o}\mathbf{d},B\text{-}\mathbf{M}\mathbf{o}\mathbf{d})$ for the schematic category of these functors and natural transformations, with componentwise identities and vertical composition. As prescribed by Functor category $[\mathcal{C},\mathcal{D}]$ and Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{C}\mathbf{A}\mathbf{T}$ is not formed, this is metatheoretic shorthand: functors on the large module category are fixed definable-class schemas, not set-coded objects of a ZFC class category. All categorical assertions here and in the Eilenberg–Watts equivalence are understood componentwise for fixed such schemas. For each fixed pair $F,G$, Natural transformations of additive cocontinuous module functors are determined at the regular module ↗ proves that a natural transformation is determined by its component at $A$ and that the admissible components form a set. We use that set as $\mathrm{Nat}(F,G)$; the corresponding proper-class families themselves are not elements of a set. No category of all definable-class functors is formed.

The term is a property of a functor; no commutativity of $A$ or $B$ is assumed, and no choice is used.