TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data

Statement

Let $U : A \to C$ , where $A$ is complete and locally small, $C$ is locally small, and $A$ has a supplied small coseparating set. Assume that $U$ preserves all small limits. Fix $C \in C$ . Suppose in addition one of the following data is supplied:

$A$ has a supplied well-powering; or
every collection of subobjects in $A$ has a specified intersection and $U$ preserves the pullbacks of the corresponding families of monomorphisms, including the possibly proper collections invoked in the proof.

Then $(C ↓ U)$ has an initial object.

Preservation of all small limits is required in both branches, not only in the first. The proof produces the initial object inside $(C ↓ U)$ from A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object, which needs $(C ↓ U)$ to be complete, and the comma projection creates only those limits that $U$ preserves. The second branch is therefore not a weakening of that hypothesis: it adds preservation data for the possibly proper collections, rather than treating such a collection as a small diagram.

Without it the conclusion fails. Take $A = C = S e t$ , let $U$ be the constant functor at the two-element set $2$ , and let $C = 1$ . Then $S e t$ is complete and locally small, ${2}$ is a small coseparating set, every collection of subobjects has its intersection, and $U$ carries each wide pullback of monomorphisms to a cone that is again a limit, since the diagram is connected and $U$ is constant; so the branch-2 data is supplied. But $U$ is not continuous — it does not preserve the empty limit — and $(1 ↓ U)$ is the disjoint union of two copies of $S e t$ , one for each map $1 \to 2$ , which has no initial object.

Facts & Assumptions

Given: The functor, fixed object, categorical hypotheses including preservation of all small limits by $U$ , and one of the two supplied branches in the Statement.

[L1]

A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object (A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object).

[L2]

A supplied well-powering gives, as data for every object $C$ at once, a set $M_{C}$ of monomorphisms into $C$ containing a representative of every subobject class of $C$ (Well-powered and co-well-powered categories, and supplied well-powerings).

[L3]

A set-indexed wide pullback computes the intersection independently of representatives (Wide pullbacks compute intersections of supplied set-indexed subobject representatives independently of the representatives).

[L4]

The comma projection strictly creates every projected limit preserved by $U$ (A comma-category projection strictly creates the limits preserved by the functor).

[L5]

A coseparating set detects distinct maps by postcomposition (Separating and coseparating sets of objects), completeness concerns all small limits (Finite, small, and large limits and colimits; complete and cocomplete categories), a functor is continuous when it preserves all small limits (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors), and local smallness makes hom-collections sets (Small, locally small, and large categories).

Proof

technique · cases

1.1L5

The comma category is locally small because its hom-collections are subsets of those in $A$ . The set of all comma objects $C \to U (K)$ with $K$ in the supplied coseparating set is again a set by local smallness of $C$ , and it is coseparating by [L5].

1.2assume-case poweredL2L3L4L5

Assume the supplied-well-powering branch. The subobjects of a fixed comma object project injectively into subobject classes of its $A$ -component: the projection preserves and reflects monomorphisms, and preservation of pullbacks makes the projected monomorphisms remain monic after applying $U$ . By [L2] they therefore admit a supplied set of representatives. Their wide pullback exists by completeness, is preserved by continuity, and [L3] and [L4] create its intersection in the comma category.

1.3assume-case directL4L5

Assume the direct-intersection branch. Intersect the projected collection using the stated class-intersection datum, including its empty-collection case, and use the separately supplied preservation of that family-of-monomorphisms pullback to construct the comma structure arrow. This invokes no proper-class diagram and does not infer that preservation from the assumed continuity, which covers only small diagrams.

2.1step 1.1step 1.2step 1.3L1L4L5givencases-exhaustive∎

In either branch, $A$ is complete and $U$ preserves all small limits by hypothesis, so [L4] creates every small limit in $(C ↓ U)$ and the comma category is complete for small diagrams. It is locally small, has the coseparating set of step 1.1, and has all the subobject intersections needed by [L1] — from the representative sets of step 1.2 in the first branch, and from the supplied class intersections of step 1.3 in the second. Hence [L1] gives an initial object of $(C ↓ U)$ .

Depends on

Used by

Dependency tree · two levels

24 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

S. Mac Lane, Categories for the Working Mathematician, theorem V.8.2 and corollary (standard reference, not scraped)
E. Riehl, Category Theory in Context, theorem 4.7.10 (standard reference, not scraped)