TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Right adjoints preserve every limit that exists

Statement

Let $F : C \to D$ be left adjoint to $G : D \to C$ . If a diagram $D : J \to D$ has a limit $(L, λ)$ , then $(G L, G λ)$ is a limit of $G D$ . Thus $G$ preserves every limit that exists, for arbitrary indexing categories for which the displayed diagram and cone categories are legitimate.

Facts & Assumptions

Given: An adjunction $F ⊣ G$ with unit $η$ and counit $ε$ , a diagram $D : J \to D$ , and a limiting cone $λ_{j} : L \to D_{j}$ .

[F1]

A limit of a diagram is a terminal cone: every cone has a unique mediating morphism to its vertex whose composites with the limiting legs are the given cone legs (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[F2]

A functor preserves a limit when the image of a limiting cone is limiting (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

[L1]

The unit and counit are natural and satisfy $ε_{F c} F (η_{c}) = 1_{F c}$ and $G (ε_{d}) η_{G d} = 1_{G d}$ (Adjunction by unit, counit, and the triangle identities).

Proof

technique · direct

1.1L1construct

Let $μ_{j} : c \to G D_{j}$ be any cone over $G D$ , and define ${\overset{ˉ}{μ}}_{j} : = ε_{D_{j}} \circ F (μ_{j}) : F c \to D_{j}$ .

2.1step 1.1L1

For $a : j \to k$ in $J$ , naturality of $ε$ and the cone equation for $μ$ give $D (a) {\overset{ˉ}{μ}}_{j} = ε_{D_{k}} F G D (a) F (μ_{j}) = ε_{D_{k}} F (G D (a) μ_{j}) = {\overset{ˉ}{μ}}_{k}$ , so $\overset{ˉ}{μ}$ is a cone over $D$ .

3.1step 2.1F1construct

By [F1], there is a unique $h : F c \to L$ with $λ_{j} h = {\overset{ˉ}{μ}}_{j}$ for every $j$ . Define $\hat{h} : = G (h) η_{c} : c \to G L$ .

4.1step 1.1step 3.1L1

For each $j$ , $G (λ_{j}) \hat{h} = G ({\overset{ˉ}{μ}}_{j}) η_{c} = G (ε_{D_{j}}) G F (μ_{j}) η_{c} = G (ε_{D_{j}}) η_{G D_{j}} μ_{j} = μ_{j}$ , by naturality of $η$ and the second triangle identity.

4.2step 1.1step 3.1F1L1

If $k : c \to G L$ also satisfies $G (λ_{j}) k = μ_{j}$ , then $ε_{L} F (k) : F c \to L$ mediates $\overset{ˉ}{μ}$ by naturality of $ε$ . Uniqueness in [F1] makes it $h$ , and naturality of $η$ together with the second triangle identity $G (ε_{L}) η_{G L} = 1_{G L}$ give $k = G (ε_{L}) G F (k) η_{c} = G (ε_{L}) η_{G L} k$ , that is $k = G (h) η_{c} = \hat{h}$ .

5.1step 4.1step 4.2F1F2∎

Thus every cone over $G D$ factors uniquely through $(G L, G λ)$ , so it is limiting by [F1]. The construction also covers the empty diagram, where there are no leg equations, and [F2] says exactly that $G$ preserves the limit.

Right adjoints preserve every limit that exists

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources