Let (F, G) be an adjoint pair, where F: 𝒞 →𝒟 and G: 𝒟 →𝒞, and let τC, D: Hom(F C, D) →Hom(C, G C

step 1 So we want to show the composition of bi -jective functions is bi -jective. So proof is suppose

Let (F, G) be an adjoint pair, where F: 𝒞 →𝒟 and G: 𝒟 →𝒞,...

Let $(F, G)$ be an adjoint pair, where $F: \mathcal{C} \rightarrow \mathcal{D}$ and $G: \mathcal{D} \rightarrow \mathcal{C}$, and let $\tau_{C, D}: \operatorname{Hom}(F C, D) \rightarrow \operatorname{Hom}(C, G C)$ be the natural bijection. (i) If $D=F C$, there is a natural bijection $$ \tau_{C, F C}: \operatorname{Hom}(F C, F C) \rightarrow \operatorname{Hom}(C, G F C) $$ with $\tau\left(1_{F C}\right)=\eta_{C}: C \rightarrow G F C$. Prove that $\eta: 1_{C} \rightarrow G F$ is a natural transformation. (ii) If $C=G D$, there is a natural bijection $$ \tau_{G D, D}^{-1}: \operatorname{Hom}(G D, G D) \rightarrow \operatorname{Hom}(F G D, D) $$ with $\tau^{-1}\left(1_{D}\right)=\varepsilon_{D}: F G D \rightarrow D .$ Prove that $\varepsilon: F G \rightarrow$ $1_{\mathcal{D}}$ is a natural transformation. (We call $\varepsilon$ the counit of the adjoint pair.)

Let $(F, G)$ be an adjoint pair, where $F: \mathcal{C} \rightarrow \mathcal{D}$ and $G: \mathcal{D} \rightarrow \mathcal{C}$, and let $\tau_{C, D}: \operatorname{Hom}(F C, D) \rightarrow \operatorname{Hom}(C, G C)$ be the natural bijection.
(i) If $D=F C$, there is a natural bijection
$$
\tau_{C, F C}: \operatorname{Hom}(F C, F C) \rightarrow \operatorname{Hom}(C, G F C)
$$
with $\tau\left(1_{F C}\right)=\eta_{C}: C \rightarrow G F C$. Prove that $\eta: 1_{C} \rightarrow G F$ is a natural transformation.
(ii) If $C=G D$, there is a natural bijection
$$
\tau_{G D, D}^{-1}: \operatorname{Hom}(G D, G D) \rightarrow \operatorname{Hom}(F G D, D)
$$
with $\tau^{-1}\left(1_{D}\right)=\varepsilon_{D}: F G D \rightarrow D .$ Prove that $\varepsilon: F G \rightarrow$ $1_{\mathcal{D}}$ is a natural transformation. (We call $\varepsilon$ the counit of the adjoint pair.)

Key Concepts

Naturality Condition

The naturality condition requires that a natural transformation intertwines with the morphisms of the categories involved so that the corresponding diagrams commute. This is a central requirement for ensuring that constructions like the unit or counit of an adjunction are consistent with the functorial actions on all objects and morphisms, maintaining coherence across the entire category.

Hom-set Natural Isomorphism

A Hom-set natural isomorphism is the establishment of a bijective relationship between hom-sets that is natural in both arguments. In the theory of adjunctions, this isomorphism is crucial because it defines the adjunction itself, linking the morphisms in one category directly with those in another in a way that respects the categorical structure.

Counit of an Adjunction

The counit of an adjunction is a natural transformation from the composite functor F ? G to the identity functor on the target category. It represents the canonical way of translating an object obtained by applying the adjoint pair back to its original form, and its naturality means that this process commutes with the morphisms in the target category.

Unit of an Adjunction

The unit of an adjunction is a natural transformation from the identity functor on the source category to the composite functor G ? F, derived from the natural bijection associated with an adjoint pair. It encapsulates the idea that each object maps naturally into the evaluation of the adjoint pair, and its naturality ensures that these mappings work coherently with the morphisms in the category.

Natural Transformation

A natural transformation is a way to transform one functor into another while respecting the internal structure of the categories involved. In this context, it ensures that the components of the transformation—each a morphism—commute with the action of the functors on all morphisms in the source category, capturing the idea of a uniformly coherent change between functors.

Adjoint Functors

Adjoint functors are a pair of functors F and G between two categories where there exists a natural bijection between the hom-sets Hom(F(C), D) and Hom(C, G(D)). This structure underpins many constructions in category theory, linking the properties of objects and morphisms across different categories in a way that preserves and reflects the categorical structure.

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