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The category $\mathbb{R}$ is defined as a poset category where the objects are real numbers, and there is a morphism $x \to y$ exactly when $x \le y$. This means the morphisms encode the ordinary order relation on the real numbers.

Conditions: Objects are real numbers.; A morphism $x\to y$ exists precisely when $x\le y$.

The category $\mathbb{R}$ is defined as a poset category where the objects are real numbers, and there is a morphism $x \to y$ exactly when $x \le y$. This means the morphisms encode the ordinary order relation on the real numbers.

Conditions: Objects are real numbers.; A morphism $x\to y$ exists precisely when $x\le y$.

Naturality in $A$ means that the adjunction correspondence is compatible with precomposition by any morphism $h:A'\to A$. Specifically, taking a morphism $A\to G(B)$, precomposing with $h$, and then translating across the adjunction gives the same result as first translating $A\to G(B)$ to $F(A)\to B$ and then precomposing with $F(h):F(A')\to F(A)$.

Conditions: Fix an object $B\in\mathcal{B}$.; The statement quantifies over all morphisms $h:A'\to A$ in $\mathcal{A}$.; The correspondence is between morphisms in $\mathcal{A}$ and morphisms in $\mathcal{B}$.

Naturality in $A$ means that the adjunction correspondence is compatible with precomposition by any morphism $h:A'\to A$. Specifically, taking a morphism $A\to G(B)$, precomposing with $h$, and then translating across the adjunction gives the same result as first translating $A\to G(B)$ to $F(A)\to B$ and then precomposing with $F(h):F(A')\to F(A)$.

Conditions: Fix an object $B\in\mathcal{B}$.; The statement quantifies over all morphisms $h:A'\to A$ in $\mathcal{A}$.; The correspondence is between morphisms in $\mathcal{A}$ and morphisms in $\mathcal{B}$.