TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Evaluation is the limit over the coslice category

Statement

Let $F : C \to E$ be a functor and let $c$ be an object of $C$ . Then the diagram

$c / C ⟶ C \overset{F}{⟶} E$

has a limit, and that limit is $F (c)$ .

The proof is direct: the identity arrow $1_{c} : c \to c$ is initial in the coslice category, so no product or equalizer computation is needed.

Facts & Assumptions

Given: A functor $F : C \to E$ and an object $c$ of $C$ .

[F1]

The coslice category $c / C$ has objects arrows $u : c \to d$ , and a morphism from $u : c \to d$ to $u^{'} : c \to d^{'}$ is an arrow $a : d \to d^{'}$ with $a \circ u = u^{'}$ (Comma category, slice category, and coslice category).

[F2]

A limit is a terminal cone (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[L1]

The comma-category formulas compute right Kan extensions (Comma-category limit and colimit formulae compute Kan extensions).

Proof

technique · direct

1.1

Specialize the right-Kan part of [L1] to $K = 1_{C}$ . Then the indexing category $(c ↓ 1_{C})$ is exactly the coslice category $c / C$ of [F1].

F1L1

2.1

The object $(c, 1_{c})$ is initial in $c / C$ : for any object $u : c \to d$ , the required morphism from $(c, 1_{c})$ to $u$ is $u$ itself, and it is unique because its defining equation is $a \circ 1_{c} = u$ . Therefore the image of $(c, 1_{c})$ under the diagram is a limit object. That image is $F (c)$ , and its limiting cone has components $F (u) : F (c) \to F (d)$ . By [F2], this says precisely that $F (c)$ is the limit of the diagram on $c / C$ .

F1F2step 1.1∎

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

E. Riehl, Category Theory in Context, 2nd ed., Proposition 6.5.4 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Proposition 4.7.2 (standard reference, not scraped)