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The ( unique ) representable functor F: → is the Cayley representation of G. In fact, this functor is isomorphic to and so sends to the set which is by definition the " set " G and the morphism g of ( i. e. the element g of G ) to the permutation F < sub > g </ sub > of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group

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## Some Related Sentences

unique and functor

Given a

__functor__U**and**an object X as above**,**there may or may not exist an initial**morphism****from**X**to**U**.**If**,**however**,**an initial**morphism****(**A**,**φ**)**does exist then it**is**essentially__unique__**.****which**

**is**natural in

**the**variable N

**.**Here

**the**

__functor__Hom

**(**N

**,**

**F**–)

**is**

**the**composition

**of**

**the**Hom

__functor__Hom

**(**N

**,**–) with

**F**

**.**This isomorphism

**is**

**the**

__unique__one

**which**respects

**the**limiting cones

**.**

For a given diagram

**F****:**J**→**C**and**__functor__**G****:**C**→**D**,**if both**F****and**GF have specified limits there**is**a__unique__canonical**morphism**
A

__functor__**G**lifts limits uniquely for a diagram**F**if there**is**a__unique__preimage cone**(**L ′, φ ′) such**that****(**L ′, φ ′)**is**a limit**of****F****and****G****(**L ′, φ ′) =**)**

**The**variety

**that**represents

**this**

__functor__

**is**called

**the**restriction

**of**scalars

**,**

**and**

**is**

__unique__up

**to**

__unique__isomorphism if it exists

**.**

**:**A universal

**element**

**of**a

__functor__

**F**

**:**C

**→**Set

**is**a pair

**(**A

**,**u

**)**consisting

**of**an object A

**of**C

**and**an

**element**u ∈

**F**

**(**A

**)**such

**that**for every pair

**(**X

**,**v

**)**with v ∈

**F**

**(**X

**)**there exists a

__unique__

**morphism**f

**:**A

**→**X such

**that**

**(**Ff

**)**u = v

**.**

*

**The**forgetful__functor__U**:**Grp**→**Set**is**faithful as each**group**maps**to**a__unique__**set****and****the****group**homomorphism are a subset**of****the**functions**.**
This

**is**enough**to**show**that**right derived functors**of**any left exact__functor__exist**and**are__unique__up**to**canonical isomorphism**.**
A

__functor__**F****:**1**→**Set maps**the**__unique__object**of**1**to**some**set**S**and****the**__unique__identity arrow**of**1**to****the**identity function 1**<****sub****>**S**</****sub****>**on S**.**A subfunctor**G****of****F**maps**the**__unique__object**of**1**to**a subset T**of**S**and**maps**the**__unique__identity arrow**to****the**identity function 1**<****sub****>**T**</****sub****>**on T**.**Notice**that**1**<****sub****>**T**</****sub****>****is****the**restriction**of**1**<****sub****>**S**</****sub****>****to**T**.**Consequently**,**subfunctors**of****F**correspond**to**subsets**of**S**.**
Formally

**,****the**right Kan extension**of**along consists**of**a__functor__**and**a natural transformation**which****is**couniversal with respect**to****the**specification**,**in**the**sense**that**for any__functor__**and**natural transformation**,**a__unique__natural transformation**is**defined**and**fits into a commutative diagram
This gives rise

**to****the**alternate description**:****the**left Kan extension**of**along consists**of**a__functor__**and**a natural transformation**which**are universal with respect**to****this**specification**,**in**the**sense**that**for any other__functor__**and**natural transformation**,**a__unique__natural transformation exists**and**fits into a commutative diagram**:**

unique and F

Namely φ

**is**universal for homomorphisms**from****G****to**an abelian**group**H**:**for any abelian**group**H**and**homomorphism**of**groups f**:****G****→**H there exists a__unique__homomorphism__F__**:****G****<**sup**>**ab**</**sup**>****→**H such**that****.**
However

**,**in principle**,**since**the**same electronegativities should be obtained for any two bonding compounds**,****the**data**is**in**fact**overdetermined**,****and****the**signs are__unique__once a reference point**is**fixed**(**usually**,**for H or__F__).
If we require

**that****the**Lie**group**be simply connected**,**then**the**global structure**is**determined**by**its Lie algebra**:**for every finite dimensional Lie algebra over__F__there**is**a simply connected Lie**group****G**with as Lie algebra**,**__unique__up**to**isomorphism**.**
If

__F__**is**a field**and**f**and****g**are polynomials in__F__with**g**≠ 0**,**then there exist__unique__polynomials q**and**r in__F__with
As Mason uses his

__unique__experience**to**escape**from**their cells**,**he reveals why he was held there for**so**many years — for stealing a microfilm**of****the**United States ' most closely guarded secrets**,**including**the**Roswell UFO incident**and****the**John__F__**.**Kennedy assassination**(**Womack revealed**this****to**Paxton**,**earlier ).
Given a class function

**G****:**V**→**V**,**there exists a__unique__transfinite sequence__F__**:**Ord**→**V**(**where Ord**is****the**class**of**all ordinals**)**such**that**
As in

**the**case**of**induction**,**we may treat different types**of**ordinals separately**:**another formulation**of**transfinite recursion**is****that**given a**set****g****<****sub****>**1**</****sub**>,**and**class functions**G****<****sub****>**2**</****sub**>,**G****<****sub****>**3**</****sub**>, there exists a__unique__function__F__**:**Ord**→**V such**that**
* For each object X in C

**,****(**__F__**(**X ), η**<****sub****>**X**</****sub**>)**is**an initial**morphism****from**X**to****G****.**That**is****,**for all f**:**X**→****G****(**Y**)**there exists a__unique__**g****:**__F__**(**X**)****→**Y for**which****the**following diagrams commute**.**
That

**is****,**for all**g****:**__F__**(**X**)****→**Y there exists a__unique__f**:**X**→****G****(**Y**)**for**which****the**following diagrams commute**.**
A limit

**of****the**diagram__F__**:**J**→**C**is**a cone**(**L**,**φ**)****to**__F__such**that**for any other cone**(**N**,**ψ**)****to**__F__there exists a__unique__**morphism**u**:**N**→**L such**that**φ**<****sub****>**X**</****sub****>**o u =
A colimit

**of**a diagram__F__**:**J**→**C**is**a co-cone**(**L**,****)****of**__F__such**that**for any other co-cone**(**N**,**ψ**)****of**__F__there exists a__unique__**morphism**u**:**L**→**N such**that**u o**<****sub****>**X**</****sub****>**= ψ**<****sub****>**X**</****sub****>**for all X in J**.**
As with limits

**,**if a diagram__F__has a colimit then**this**colimit**is**__unique__up**to**a__unique__isomorphism**.****which**assigns each diagram its limit

**and**each natural transformation η

**:**

__F__

**→**

**G**

**the**

__unique__

**morphism**lim η

**:**lim

__F__

**→**lim

**G**commuting with

**the**corresponding universal cones

**.**

unique and →

If K

**is**a subset**of**ker**(**f**)**then there exists a__unique__homomorphism h**:****G**/ K__→__H such**that**f = h φ**.**
If it does

**,**however**,**it**is**__unique__in a strong sense**:**given any other inverse limit X ′ there exists a__unique__isomorphism X ′__→__X commuting with**the**projection maps**.****The**homomorphism η

**is**characterized

**by**

**the**following universal property

**:**given any profinite

**group**H

**and**any

**group**homomorphism f

**:**

**G**

__→__H

**,**there exists a

__unique__continuous

**group**homomorphism

**g**

**:**

**G**

**<**sup >^</ sup

**>**

__→__H with f

* Whenever Y

**is**an object**of**D**and**f**:**X__→__U**(**Y**)****is**a**morphism**in C**,**then there exists a__unique__**morphism****g****:**A__→__Y such**that****the**following diagram commutes**:**
* Whenever Y

**is**an object**of**D**and**f**:**U**(**Y**)**__→__X**is**a**morphism**in C**,**then there exists a__unique__**morphism****g****:**Y__→__A such**that****the**following diagram commutes**:**
such

**that**for any other object Z**of**D**and**morphisms f**:**Z__→__X**and****g****:**Z__→__Y there exists a__unique__**morphism**h**:**Z__→__X × Y such**that**f
Specifically

**,**it**is**__unique__up**to**a__unique__isomorphism**:**if**(**A ′, φ ′)**is**another such pair**,**then there exists a__unique__isomorphism k**:**A__→__A ′ such**that**φ ′
Suppose

**(**A**<****sub****>**1**</****sub**>, φ**<****sub****>**1**</****sub**>)**is**an initial**morphism****from**X**<****sub****>**1**</****sub****>****to**U**and****(**A**<****sub****>**2**</****sub**>, φ**<****sub****>**2**</****sub**>)**is**an initial**morphism****from**X**<****sub****>**2**</****sub****>****to**U**.**By**the**initial property**,**given any**morphism**h**:**X**<****sub****>**1**</****sub****>**__→__X**<****sub****>**2**</****sub****>**there exists a__unique__**morphism****g****:**A**<****sub****>**1**</****sub****>**__→__A**<****sub****>**2**</****sub****>**such**that****the**following diagram commutes**:**

unique and is

What makes

**the**current phenomenon__unique____is__**that****so**many science-fiction writers have reversed a trend**and**turned**to**writing works critical**of****the**impact**of**science**and**technology on human life**.**
One

**of****the**inescapable realities**of****the**Cold War__is__**that**it has thrust upon**the**West a wholly new**and**historically__unique__**set****of**moral dilemmas**.**
A number

**of**__unique__medical problems might be created when man__is__exposed**to**an infectious agent through**the**respiratory route rather than**by**natural portal**of**entry**.**
It

__is__interesting**that**a 1**:**1 correspondence can be established between**the**lines**of**two such pencils**,****so****that**in a sense a__unique__image can actually be assigned**to**each tangent**.**
Hence

**,**thought**of**as a line in a particular plane **yp**,**any tangent**to**Q has a__unique__image**and**moreover**this**image__is__**the**same for all planes through L**.**
This weakness

__is__not__unique__**to**labor surplus areas**,**for it__is__inherent in**the**system**of**local school districts in**this**country**.**
What with traders trading for

**so**many different objectives**,****and**what with there being**so**many__unique__**and**individualized market theories**and**trading techniques in use**,****and**more coming into use all**the**time**,**it__is__hard**to**imagine how any particular theory or technique could acquire enough `` fans ''**to**invalidate itself**.**
Probably

**the**primary reason for special treatment**of**a net operating loss carryover__is__**the**__unique__opportunity it presents for tax avoidance**.**
It

__is__**the**classroom teacher**,**however**,**who has daily contacts with pupils**,****and**who__is__in a__unique__position**to**put sound psychological principles into practice**.****The**policy may not be

__unique__but

**the**maximum value

**of**P certainly

__is__

**,**

**and**once

**the**policy

__is__specified

**this**maximum can be calculated

**by**

**(**2

**)**

**and**

**(**3

**)**as a function

**of**

**the**feed state Af

**.**

Sir Julian Huxley in his book Uniqueness Of Man makes

**the**novel point**that**just as man__is____unique__in being**the**only animal**which**requires a long period**of**infancy**and**childhood under family protection**,****so**__is__he**the**only animal who has a long period after**the**decline**of**his procreativity**.**
Most people do not realize

**that****the**congregation**,**as a gathered fellowship meeting regularly face**to**face**,**personally sharing in a common experience**and**expressing**that**experience in daily relationships with one another**,**__is____unique__**.**
A sense

**of**self-certainty**and****the**freedom**to**experiment with different roles**,**or confidence in one's own__unique__behavior as an alternative**to**peer-group conformity**,**__is__more easily developed during adolescence if**,**during early childhood**,****the**individual was permitted**to**exercise initiative**and**encouraged**to**develop some autonomy**.**0.295 seconds.