TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The internal hom preserves limits in the covariant variable and sends colimits to limits in the contravariant variable

Statement

In a biclosed monoidal category, for each fixed object $X$ the functor $[X, -]$ preserves all limits that exist. For each fixed object $Y$ , the contravariant functor $[-, Y] : C^{o p} \to C$ preserves limits in $C^{o p}$ ; equivalently, it sends colimits in $C$ to limits in $C$ .

Facts & Assumptions

Given: A biclosed monoidal category.

[L1]

For each fixed $X$ , the right internal hom $[X, -]$ is right adjoint to $- \otimes X$ (The internal hom and its evaluation morphism).

[L2]

Right adjoints preserve all limits that exist in their domain (Right adjoints preserve every limit that exists).

[L3]

In a biclosed monoidal category, tensoring with a fixed object preserves every colimit that exists (In a biclosed monoidal category tensor is cocontinuous in each variable).

Proof

technique · direct

1.1givenL1L2

Fix $X$ . By [L1], the functor $[X, -]$ is a right adjoint, so [L2] implies that it preserves every limit that exists in $C$ .

1.2

Fix $Y$ . For a morphism $u : X^{'} \to X$ , define $[u, Y] : [X, Y] \to [X^{'}, Y]$ to be the transpose of

$[X, Y] \otimes X^{'} \overset{1 \otimes u}{\to} [X, Y] \otimes X \overset{{ev}_{X, Y}}{\to} Y .$

The transposition bijection of [L1] makes this assignment contravariantly functorial in $X$ . [given, L1, construct]

2.1givenstep 1.2L3algebra

Let $(i_{j} : X_{j} \to X)$ be a colimit cocone in $C$ , and fix an object $A$ . By [L3], the functor $A \otimes -$ preserves this colimit, so maps $A \otimes X \to Y$ are naturally the same as compatible families of maps $A \otimes X_{j} \to Y$ .

3.1step 1.2step 2.1L1algebra

Apply the transposition bijection of [L1] to the maps in step 2.1. A map $A \otimes X \to Y$ is the same as a map $A \to [X, Y]$ , and a compatible family $A \otimes X_{j} \to Y$ is the same as a compatible family $A \to [X_{j}, Y]$ with respect to the morphisms $[i_{j}, Y]$ from step 1.2. Therefore maps $A \to [X, Y]$ are naturally in bijection with cones from $A$ to the diagram $j \mapsto [X_{j}, Y]$ , so $[X, Y]$ is a limit of that diagram.

4.1step 1.1step 3.1algebra∎

Hence the contravariant functor $[-, Y] : C^{o p} \to C$ sends colimits in $C$ to limits in $C$ , equivalently preserves limits in $C^{o p}$ . Combining this with step 1.1 proves the claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Theorem 4.2.1 and Section 4.4 (standard reference, not scraped)