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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 $\mathcal{B}$.

The symbol $\cong$ denotes an isomorphism in the category of sets, which means it is simply a bijection between the two hom-sets. Because hom-sets are sets, their isomorphisms are bijective functions.

Conditions: The compared objects are hom-sets; The ambient category for the isomorphism is the category of sets

The symbol $\cong$ denotes an isomorphism in the category of sets, which means it is simply a bijection between the two hom-sets. Because hom-sets are sets, their isomorphisms are bijective functions.

Conditions: The compared objects are hom-sets; The ambient category for the isomorphism is the category of sets