Properties of adjoints

We list the following two properties of adjoints without proof.

Uniqueness of adjoints

Any two right adjoints of a given functor $F : C \to D$ are naturally isomorphic, as are any two left adjoints of a functor $G : D \to C$ .

Adjoints and (co)limits

Suppose $F : C \to D$ is a functor and $T : J \to C$ is a diagram of shape $J$ in $C .$

If $F$ is a left adjoint (of some functor $G : D \to C$ ) and if the diagram $T$ has a colimiting cone $τ : T \Rightarrow c$ in $C$ , then $F T$ has the colimiting cone $F τ : F (c) \Rightarrow F T$ in $D$ .

If $F$ is right adjoint (of some functor $G : D \to C$ ), and if the diagram $T$ has a limiting cone $τ : c \Rightarrow T$ in $C$ , then $F T$ has the limiting cone $F τ : F T \Rightarrow F (c)$ in $D$ .

In short, left adjoints preserve colimits (that exist) and right adjoints preserve limits (that exist).

Examples of adjoints preserving (co)limits

The free $R$ -module functor $F : Set \to R -Mod$ is left adjoint to the forgetful functor $U : R -Mod \to Set$ , so it commutes with colimits. In particular, it commutes with coproducts, which in the category $Set$ is disjoint union and in the category $R -Mod$ is direct sum. Thus, for any family of sets ${S_{i} ∣ i \in I}$ we have an isomorphism

F (\underset{i \in I}{∐} S_{i}) ≃ ⨁_{i \in I} F (S_{i}) .

The tensor product functor $R \otimes_{z} - : Ab \to R -Mod$ is left adjoint to the corresponding forgetful functor, so it also commutes with colimits (and hence coproducts). The coproduct in the category $Ab$ is the direct sum, so for any family of abelian groups ${G_{i} ∣ i \in I}$ we have an isomorphism

R \otimes_{Z} (⨁_{i \in I} G_{i}) ≃ ⨁_{i \in I} (R \otimes_{Z} G_{i}) .

The hom functor ${Hom}_{Ab} (U (R), -) : Ab \to R -Mod$ is right adjoint to the corresponding forgetful functor, so it commutes with limits (and hence products). So for any family of abelian groups as above we also have an isomorphism

{Hom}_{Ab} (U (R), \prod_{i \in I} G_{i}) ≃ \prod_{i \in I} {Hom}_{Ab} (U (R), G_{i}) .

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