LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A functor preserves a chosen limit exactly when its canonical comparison to the chosen target limit is an isomorphism, and dually for colimits

Statement

Let $(L, λ)$ be a chosen limit of $D : J \to C$ , and let $(M, μ)$ be a chosen limit of $F D$ in $D$ . There is a unique canonical comparison $c : F (L) \to M$ satisfying $μ_{j} c = F (λ_{j})$ . The functor $F$ preserves this limit if and only if $c$ is an isomorphism. Dually, the canonical map from a chosen colimit of $F D$ to the image of a chosen colimit of $D$ is an isomorphism exactly when that colimit is preserved.

Facts & Assumptions

Given: The two chosen limiting cones in the statement.

[F1]

Preservation means that the image of the source limiting cone is limiting (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

[L1]

Two limits of one diagram have a unique compatible isomorphism (Any two limits, or any two colimits, of one diagram are uniquely isomorphic compatibly with their structure maps).

[L2]

Limits and colimits are formal duals (A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category).

Proof

technique · biconditional

1.1

The family $F (λ_{j})$ is a cone over $F D$ , so the universal property of $(M, μ)$ gives a unique $c : F (L) \to M$ with $μ_{j} c = F (λ_{j})$ .

given

2.1

If $F$ preserves the limit, [F1] makes $(F (L), F λ)$ another limit of $F D$ . By [L1], its unique compatible map $c$ to $(M, μ)$ is an isomorphism.

F1L1step 1.1

2.2

Conversely, suppose $c$ is an isomorphism. For any cone $ξ$ over $F D$ , its unique factor $h : X \to M$ through $μ$ yields $c^{- 1} h : X \to F (L)$ , and $F (λ_{j}) c^{- 1} h = μ_{j} h = ξ_{j}$ .

givenstep 1.1

3.1

If $r : X \to F (L)$ has these equations, then $c r$ factors $ξ$ through $μ$ , so $c r = h$ and $r = c^{- 1} h$ . Thus $F λ$ is limiting, and [F1] says that $F$ preserves this limit.

F1step 2.2

4.1

Reversing arrows in steps 1.1, 2.1, 2.2, and 3.1 by [L2] gives the colimit comparison and proves both directions of its criterion.

L2step 1.1step 2.1step 2.2step 3.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 8 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

E. Riehl, Category Theory in Context, Definition 3.4.1 (standard reference, not scraped)