CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Left adjoints preserve every colimit that exists

Statement

If $F : C \to D$ is a left adjoint and a diagram $D : J \to C$ has a colimit, then applying $F$ to a colimiting cocone produces a colimit of $F D$ . Thus left adjoints preserve every colimit that exists.

Facts & Assumptions

Given: An adjunction $F ⊣ G$ and a diagram in $C$ with a colimit.

[F1]

A formal theorem derived from the category axioms has a formal dual obtained by reversing morphisms and exchanging each notion with its opposite-category version (Every theorem about categories has a formal dual obtained by reversing morphisms and composition).

[F2]

A cocone is colimiting in a category exactly when the reversed family is a limiting cone in the opposite category (A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category).

[L1]

Right adjoints preserve every limit that exists (Right adjoints preserve every limit that exists).

Proof

technique · direct

1.1

Passing to opposite categories turns $F ⊣ G$ into $G^{o p} ⊣ F^{o p}$ , so $F^{o p}$ is a right adjoint.

1.2

By [F2], the given colimit in $C$ is a limit in $C^{o p}$ .

2.1

Apply [L1] to $F^{o p}$ : the reversed image cone is limiting in $D^{o p}$ .

step 1.1step 1.2L1

3.1

Translating back with [F2], the image cocone under $F$ is colimiting in $D$ .

step 2.1F2∎

Depends on

Used by

A right adjoint is left exact and a left adjoint is right exact Corollary
Right adjoints preserve monomorphisms and left adjoints preserve epimorphisms Corollary
Every functor with a left adjoint also has a right adjoint False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Theorem 4.6.2 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Section 6.3 (standard reference, not scraped)