Category Theory Through the Lens of Relational Ontology: 4 Natural Transformations as Modulations of Relational Alignment

If functors articulate how one systemic horizon becomes legible within another, then natural transformations articulate something subtler:

how different translations of the same horizon modulate one another without collapsing their distinct perspectives.

A natural transformation is often introduced as a “morphism between functors,” but this phrasing is misleading. It risks treating functors as inert entities and the transformation as a simple arrow between them. From a relational-ontological stance, the functor is already an active construal, a cross-horizon alignment of relational potentials.

A natural transformation, then, is a second-order alignment:

a modulation between modes of alignment, a relational tuning of how one horizon is re-situated in another.

It is not structure-preserving; it is perspective-preserving.

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1. Functors Revisited: Competing Alignments of the Same Horizon

Let $F$ and $G$ be functors from $\mathcal{C}$ to $\mathcal{D}$.
In Post 3, we framed a functor as a cross-scale interpreter that positions the horizon of $\mathcal{C}$ within $\mathcal{D}$.

With two such interpreters in place, we are no longer dealing with a single translation but with two distinct construals of how $\mathcal{C}$’s potential appears when refracted through $\mathcal{D}$’s systemic space.

The question becomes:
How can we articulate the relational difference between these construals without dissolving one into the other?

Natural transformations are the answer.

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2. The Component: A Local Re-Cut of Potential

A natural transformation $\alpha: F \Rightarrow G$ assigns to each object $X$ of $\mathcal{C}$ a morphism

$${\alpha }_X:F(X)\to G(X)$$

in $\mathcal{D}$.

Conventionally, this is taught as “each component gives a way to go from $F(X)$ to $G(X)$.”
But in relational terms, the component is something much more precise:

It is a local re-cut of potential.
It tells us how the construal enacted by $F$ at the point $X$ can be modulated—without breaking coherence—into the construal enacted by $G$.

The component is not a connection between objects; it is a connection between perspectives on systemic potential.

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3. The Naturality Condition as Coherence of Construal

The naturality square is usually depicted as a commutative diagram. Abstractly:

$$G(f)\circ {\alpha }_X={\alpha }_Y\circ F(f)$$

In relational terms, this expresses a much more potent idea:

The modulation between perspectives at each local cut must itself align with how those horizons are traversed.

If functors articulate two ways of “travelling” through $\mathcal{C}$’s relational horizon inside $\mathcal{D}$, then naturality ensures that the modulation between them is path-neutral.

The transformation does not depend on which route you take through the horizon; it depends only on the systemic role of the object you began with.

This is precisely why the diagram must commute: the modulation must be a coherent adjustment of construal, not an arbitrary jump between incompatible translational stances.

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4. Natural Transformations as Second-Order Alignment

A natural transformation is thus a meta-alignment:

• Functors are alignments between horizons;

• Natural transformations are alignments between alignments.

This second-order nature matters. It means that natural transformations operate at the level of relations between construals, not at the level of objects or morphisms directly.

They articulate how two distinct ways of interpreting relational potential in $\mathcal{C}$ remain commensurable within $\mathcal{D}$.

This is the category-theoretic analogue of what, in relational ontology, we would call an intra-systemic modulation of perspective: the way two interpretive stances can differ yet remain mutually intelligible within a higher-order horizon.

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5. The Significance for Relational Ontology

Relational ontology is committed to meaning as constituted by construal, with no unconstrued phenomenon beneath it.

Given this, we must understand not only:

• how one horizon construes another (functors),
  but also:

• how different construals of the same horizon can themselves enter into relationship.

This is where natural transformations become philosophically indispensable.

They model:

• the plasticity of interpretation,

• the stability of systemic alignment,

• and the conditions under which distinct construals remain comparable.

They show how meaning can differ without fragmenting, and how systemic potential can be reconstrualed without losing coherence.

Natural transformations thus articulate the infrastructure of co-individuation: how multiple interpretive stances on the same domain can be collectively sustained, modulated, and navigated without collapsing horizons.

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6. Toward Post 5: Higher-Order Relationality

The movement from categories (first-order horizons), to functors (cross-horizon alignments), to natural transformations (modulations of alignment) lays the groundwork for a further move:

the emergence of higher-order relational spaces, where natural transformations themselves become the objects and morphisms of a new horizon.

In categorical terms: the rise of functor categories, 2-categories, and beyond.

In relational-ontological terms: the stratification of perspectives on perspectives, the deepening of how relational potential can be actualised across levels.

This is the focus of Post 5.