Definition df-up 50383

Description: Definition of the class of universal properties.

Given categories 𝐷 and 𝐸, if 𝐹:𝐷⟶𝐸 is a functor and 𝑊 an object of 𝐸, a universal pair from 𝑊 to 𝐹 is a pair ⟨𝑋, 𝑀⟩ consisting of an object 𝑋 of 𝐷 and a morphism 𝑀:𝑊⟶𝐹𝑋 of 𝐸, such that to every pair ⟨𝑦, 𝑔⟩ with 𝑦 an object of 𝐷 and 𝑔:𝑊⟶𝐹𝑦 a morphism of 𝐸, there is a unique morphism 𝑘:𝑋⟶𝑦 of 𝐷 with 𝐹𝑘 ⚬ 𝑀 = 𝑔. Such property is commonly referred to as a universal property. In our definition, it is denoted as 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀.

Note that the universal pair is termed differently as "universal arrow" in p. 55 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf (retrieved 6 Oct 2025). Interestingly, the "universal arrow" is referring to the morphism 𝑀 instead of the pair near the end of the same piece of the text, causing name collision. The name "universal arrow" is also adopted in papers such as https://arxiv.org/pdf/2212.08981. Alternatively, the universal pair is called the "universal morphism" in Wikipedia (https://en.wikipedia.org/wiki/Universal_property) as well as published works, e.g., https://arxiv.org/pdf/2412.12179. But the pair ⟨𝑋, 𝑀⟩ should be named differently as the morphism 𝑀, and thus we call 𝑋 the universal object, 𝑀 the universal morphism, and ⟨𝑋, 𝑀⟩ the universal pair.

Given its existence, such universal pair is essentially unique (upeu3 50404), and can be generated from an existing universal pair by isomorphisms (upeu4 50405). See also oppcup 50416 for the dual concept.

(Contributed by Zhi Wang, 24-Sep-2025.)

Assertion

Ref	Expression
df-up	⊢ UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))

Distinct variable group:   𝑏,𝑐,𝑑,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘,𝑚,𝑜,𝑤,𝑥,𝑦

Detailed syntax breakdown of Definition df-up

Step	Hyp	Ref	Expression
1		cup 50382	. 2 class UP
2		vd	. . 3 setvar 𝑑
3		ve	. . 3 setvar 𝑒
4		cvv 3450	. . 3 class V
5		vb	. . . 4 setvar 𝑏
6	2	cv 1568	. . . . 5 class 𝑑
7		cbs 17392	. . . . 5 class Base
8	6, 7	cfv 6538	. . . 4 class (Base‘𝑑)
9		vc	. . . . 5 setvar 𝑐
10	3	cv 1568	. . . . . 6 class 𝑒
11	10, 7	cfv 6538	. . . . 5 class (Base‘𝑒)
12		vh	. . . . . 6 setvar ℎ
13		chom 17444	. . . . . . 7 class Hom
14	6, 13	cfv 6538	. . . . . 6 class (Hom ‘𝑑)
15		vj	. . . . . . 7 setvar 𝑗
16	10, 13	cfv 6538	. . . . . . 7 class (Hom ‘𝑒)
17		vo	. . . . . . . 8 setvar 𝑜
18		cco 17445	. . . . . . . . 9 class comp
19	10, 18	cfv 6538	. . . . . . . 8 class (comp‘𝑒)
20		vf	. . . . . . . . 9 setvar 𝑓
21		vw	. . . . . . . . 9 setvar 𝑤
22		cfunc 18034	. . . . . . . . . 10 class Func
23	6, 10, 22	co 7420	. . . . . . . . 9 class (𝑑 Func 𝑒)
24	9	cv 1568	. . . . . . . . 9 class 𝑐
25		vx	. . . . . . . . . . . . 13 setvar 𝑥
26	25, 5	wel 2145	. . . . . . . . . . . 12 wff 𝑥 ∈ 𝑏
27		vm	. . . . . . . . . . . . . 14 setvar 𝑚
28	27	cv 1568	. . . . . . . . . . . . 13 class 𝑚
29	21	cv 1568	. . . . . . . . . . . . . 14 class 𝑤
30	25	cv 1568	. . . . . . . . . . . . . . 15 class 𝑥
31	20	cv 1568	. . . . . . . . . . . . . . . 16 class 𝑓
32		c1st 7999	. . . . . . . . . . . . . . . 16 class 1st
33	31, 32	cfv 6538	. . . . . . . . . . . . . . 15 class (1st ‘𝑓)
34	30, 33	cfv 6538	. . . . . . . . . . . . . 14 class ((1st ‘𝑓)‘𝑥)
35	15	cv 1568	. . . . . . . . . . . . . 14 class 𝑗
36	29, 34, 35	co 7420	. . . . . . . . . . . . 13 class (𝑤𝑗((1st ‘𝑓)‘𝑥))
37	28, 36	wcel 2144	. . . . . . . . . . . 12 wff 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))
38	26, 37	wa 401	. . . . . . . . . . 11 wff (𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥)))
39		vg	. . . . . . . . . . . . . . . 16 setvar 𝑔
40	39	cv 1568	. . . . . . . . . . . . . . 15 class 𝑔
41		vk	. . . . . . . . . . . . . . . . . 18 setvar 𝑘
42	41	cv 1568	. . . . . . . . . . . . . . . . 17 class 𝑘
43		vy	. . . . . . . . . . . . . . . . . . 19 setvar 𝑦
44	43	cv 1568	. . . . . . . . . . . . . . . . . 18 class 𝑦
45		c2nd 8000	. . . . . . . . . . . . . . . . . . 19 class 2nd
46	31, 45	cfv 6538	. . . . . . . . . . . . . . . . . 18 class (2nd ‘𝑓)
47	30, 44, 46	co 7420	. . . . . . . . . . . . . . . . 17 class (𝑥(2nd ‘𝑓)𝑦)
48	42, 47	cfv 6538	. . . . . . . . . . . . . . . 16 class ((𝑥(2nd ‘𝑓)𝑦)‘𝑘)
49	29, 34	cop 4834	. . . . . . . . . . . . . . . . 17 class ⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩
50	44, 33	cfv 6538	. . . . . . . . . . . . . . . . 17 class ((1st ‘𝑓)‘𝑦)
51	17	cv 1568	. . . . . . . . . . . . . . . . 17 class 𝑜
52	49, 50, 51	co 7420	. . . . . . . . . . . . . . . 16 class (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))
53	48, 28, 52	co 7420	. . . . . . . . . . . . . . 15 class (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)
54	40, 53	wceq 1569	. . . . . . . . . . . . . 14 wff 𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)
55	12	cv 1568	. . . . . . . . . . . . . . 15 class ℎ
56	30, 44, 55	co 7420	. . . . . . . . . . . . . 14 class (𝑥ℎ𝑦)
57	54, 41, 56	wreu 3363	. . . . . . . . . . . . 13 wff ∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)
58	29, 50, 35	co 7420	. . . . . . . . . . . . 13 class (𝑤𝑗((1st ‘𝑓)‘𝑦))
59	57, 39, 58	wral 3076	. . . . . . . . . . . 12 wff ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)
60	5	cv 1568	. . . . . . . . . . . 12 class 𝑏
61	59, 43, 60	wral 3076	. . . . . . . . . . 11 wff ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)
62	38, 61	wa 401	. . . . . . . . . 10 wff ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))
63	62, 25, 27	copab 5166	. . . . . . . . 9 class {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}
64	20, 21, 23, 24, 63	cmpo 7422	. . . . . . . 8 class (𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))})
65	17, 19, 64	csb 3846	. . . . . . 7 class ⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))})
66	15, 16, 65	csb 3846	. . . . . 6 class ⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))})
67	12, 14, 66	csb 3846	. . . . 5 class ⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))})
68	9, 11, 67	csb 3846	. . . 4 class ⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))})
69	5, 8, 68	csb 3846	. . 3 class ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))})
70	2, 3, 4, 4, 69	cmpo 7422	. 2 class (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))
71	1, 70	wceq 1569	1 wff UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))

This definition is used by:  reldmup  50384  upfval  50385