Definition df-up 49803

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 49824), and can be generated from an existing universal pair by isomorphisms (upeu4 49825). See also oppcup 49836 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 49802	. 2 class UP
2		vd	. . 3 setvar 𝑑
3		ve	. . 3 setvar 𝑒
4		cvv 3457	. . 3 class V
5		vb	. . . 4 setvar 𝑏
6	2	cv 1562	. . . . 5 class 𝑑
7		cbs 17259	. . . . 5 class Base
8	6, 7	cfv 6525	. . . 4 class (Base‘𝑑)
9		vc	. . . . 5 setvar 𝑐
10	3	cv 1562	. . . . . 6 class 𝑒
11	10, 7	cfv 6525	. . . . 5 class (Base‘𝑒)
12		vh	. . . . . 6 setvar ℎ
13		chom 17311	. . . . . . 7 class Hom
14	6, 13	cfv 6525	. . . . . 6 class (Hom ‘𝑑)
15		vj	. . . . . . 7 setvar 𝑗
16	10, 13	cfv 6525	. . . . . . 7 class (Hom ‘𝑒)
17		vo	. . . . . . . 8 setvar 𝑜
18		cco 17312	. . . . . . . . 9 class comp
19	10, 18	cfv 6525	. . . . . . . 8 class (comp‘𝑒)
20		vf	. . . . . . . . 9 setvar 𝑓
21		vw	. . . . . . . . 9 setvar 𝑤
22		cfunc 17901	. . . . . . . . . 10 class Func
23	6, 10, 22	co 7400	. . . . . . . . 9 class (𝑑 Func 𝑒)
24	9	cv 1562	. . . . . . . . 9 class 𝑐
25		vx	. . . . . . . . . . . . 13 setvar 𝑥
26	25, 5	wel 2146	. . . . . . . . . . . 12 wff 𝑥 ∈ 𝑏
27		vm	. . . . . . . . . . . . . 14 setvar 𝑚
28	27	cv 1562	. . . . . . . . . . . . 13 class 𝑚
29	21	cv 1562	. . . . . . . . . . . . . 14 class 𝑤
30	25	cv 1562	. . . . . . . . . . . . . . 15 class 𝑥
31	20	cv 1562	. . . . . . . . . . . . . . . 16 class 𝑓
32		c1st 7972	. . . . . . . . . . . . . . . 16 class 1st
33	31, 32	cfv 6525	. . . . . . . . . . . . . . 15 class (1st ‘𝑓)
34	30, 33	cfv 6525	. . . . . . . . . . . . . 14 class ((1st ‘𝑓)‘𝑥)
35	15	cv 1562	. . . . . . . . . . . . . 14 class 𝑗
36	29, 34, 35	co 7400	. . . . . . . . . . . . 13 class (𝑤𝑗((1st ‘𝑓)‘𝑥))
37	28, 36	wcel 2145	. . . . . . . . . . . 12 wff 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))
38	26, 37	wa 400	. . . . . . . . . . 11 wff (𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥)))
39		vg	. . . . . . . . . . . . . . . 16 setvar 𝑔
40	39	cv 1562	. . . . . . . . . . . . . . 15 class 𝑔
41		vk	. . . . . . . . . . . . . . . . . 18 setvar 𝑘
42	41	cv 1562	. . . . . . . . . . . . . . . . 17 class 𝑘
43		vy	. . . . . . . . . . . . . . . . . . 19 setvar 𝑦
44	43	cv 1562	. . . . . . . . . . . . . . . . . 18 class 𝑦
45		c2nd 7973	. . . . . . . . . . . . . . . . . . 19 class 2nd
46	31, 45	cfv 6525	. . . . . . . . . . . . . . . . . 18 class (2nd ‘𝑓)
47	30, 44, 46	co 7400	. . . . . . . . . . . . . . . . 17 class (𝑥(2nd ‘𝑓)𝑦)
48	42, 47	cfv 6525	. . . . . . . . . . . . . . . 16 class ((𝑥(2nd ‘𝑓)𝑦)‘𝑘)
49	29, 34	cop 4591	. . . . . . . . . . . . . . . . 17 class ⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩
50	44, 33	cfv 6525	. . . . . . . . . . . . . . . . 17 class ((1st ‘𝑓)‘𝑦)
51	17	cv 1562	. . . . . . . . . . . . . . . . 17 class 𝑜
52	49, 50, 51	co 7400	. . . . . . . . . . . . . . . 16 class (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))
53	48, 28, 52	co 7400	. . . . . . . . . . . . . . 15 class (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)
54	40, 53	wceq 1563	. . . . . . . . . . . . . 14 wff 𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)
55	12	cv 1562	. . . . . . . . . . . . . . 15 class ℎ
56	30, 44, 55	co 7400	. . . . . . . . . . . . . 14 class (𝑥ℎ𝑦)
57	54, 41, 56	wreu 3368	. . . . . . . . . . . . 13 wff ∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)
58	29, 50, 35	co 7400	. . . . . . . . . . . . 13 class (𝑤𝑗((1st ‘𝑓)‘𝑦))
59	57, 39, 58	wral 3079	. . . . . . . . . . . 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 1562	. . . . . . . . . . . 12 class 𝑏
61	59, 43, 60	wral 3079	. . . . . . . . . . 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 400	. . . . . . . . . 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 5167	. . . . . . . . 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 7402	. . . . . . . 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 3855	. . . . . . 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 3855	. . . . . 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 3855	. . . . 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 3855	. . . 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 3855	. . 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 7402	. 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 1563	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 referenced by:  reldmup  49804  upfval  49805