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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
:

where
is the unique functor from to
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
.

But Oakwood Heights
is unique in one particular
.
The structure appears
to be
unique among OOH compounds
, but
is the same as
that assumed
by Af
.

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
.

These operators D
and N are
unique and each
is a polynomial in T
.

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
.

Spontaneity training theory
is unique and relatively new
.

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
.

One reason for
the unique vitality
of the chorus
is its great variety in expression
.
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
.

First
, the State Department
is unique among government agencies for its lack
of public supporters
.
The death
of a man
is unique, and yet it
is universal
.
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