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Formally, given two categories C and D, an equivalence of categories consists of a functor F: C → D, a functor G: D → C, and two natural isomorphisms ε: FG → I < sub > D </ sub > and η: I < sub > C </ sub >→ GF.

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## Some Related Sentences

Formally and given

__Formally__

__given__to bishop Philip

**of**Senj

**,**the permission to use the Glagolitic liturgy ( the Roman Rite conducted in Slavic language instead

**of**Latin

**,**not the Byzantine rite ), actually extended to all Croatian lands

**,**mostly along the Adriatic coast

**.**

__Formally__

**,**we are

__given__

**a**set

**of**hypotheses

**and**

**a**set

**of**manifestations ; they are related by the domain knowledge

**,**represented by

**a**function that takes as

**an**argument

**a**set

**of**hypotheses

**and**gives as

**a**result the corresponding set

**of**manifestations

**.**

__Formally__

**,**the statement that " value decreases over time " is

__given__by defining the linear differential operator as

**:**

__Formally__

**,**it is

**a**norm defined on the space

**of**bounded linear operators between

**two**

__given__normed vector spaces

**.**

__Formally__

**,**the problem can be stated as follows

**:**

__given__

**a**desired property

**,**expressed as

**a**temporal logic formula p

**,**

**and**

**a**structure M with initial state s

**,**decide if

**.**

__Formally__

**,**

**an**absolute coequalizer

**of**

**a**pair in

**a**category

**C**is

**a**coequalizer as defined above but with the added property that

__given__any

**functor**

**F**( Q ) together with

**F**( q ) is the coequalizer

**of**

**F**( f )

**and**

**F**( g ) in the category

**D**

**.**Split coequalizers are examples

**of**absolute coequalizers

**.**

__Formally__

**,**

__given__

**a**partially ordered set ( P

**,**≤), then

**an**element g

**of**

**a**subset S

**of**P is the greatest element

**of**S if

__Formally__

**,**

**a**deterministic algorithm computes

**a**mathematical function ;

**a**function has

**a**unique value for any

__given__input

**,**

**and**the algorithm is

**a**process that produces this particular value as output

**.**

__Formally__

**,**

__given__

**two**partially ordered sets ( S

**,**≤)

**and**( T

**,**≤),

**a**function f

**:**S

**→**T is

**an**order-embedding if f is both order-preserving

**and**order-reflecting

**,**i

**.**e

**.**for all x

**and**y in S

**,**one has

__Formally__

**,**

__given__

**a**finite set X

**,**

**a**collection

**C**

**of**subsets

**of**X

**,**all

**of**size n

**,**has Property B if we can partition X into

**two**disjoint subsets Y

**and**Z such that every set in

**C**meets both Y

**and**Z

**.**

__Formally__

**,**

__given__

**a**graph

**G**

**,**

**a**vertex labeling is

**a**function mapping vertices

**of**

**G**to

**a**set

**of**labels

**.**

__Formally__this mission was declared to be only in pursuit

**of**the Russian vessels

**and**ammunition taken to Anzali by the White Russian counter-revolutionary General Denikin

**,**who had been

__given__asylum by British forces in Anzali

**.**

__Formally__

**,**

**a**married or widowed woman can be called by the

__given__name

**of**her husband ( Madame (

__given__name

**of**husband ) family name or Madame veuve (

__given__name

**of**husband ) family name ); this is now slightly out

**of**fashion

**.**

Formally and two

__Formally__

**,**the Congress serves

__two__functions

**:**to approve changes to the Party constitution regarding policy

**and**to elect

**a**Central Committee

**,**about 300 strong

**.**

__Formally__

**,**these failed when they were rejected by the Church

**of**England's General Synod in 1972 ; conversations

**and**co-operation continued

**,**however

**,**leading in 2003 to the signing

**of**

**a**covenant between the

__two__churches

**.**

__Formally__

**,**we start with

**a**category

**C**with finite products ( i

**.**e

**.**

**C**has

**a**terminal object 1

**and**any

__two__objects

**of**

**C**have

**a**product ).

__Formally__

**,**this means symmetry under

**a**sub-group

**of**the Euclidean group

**of**isometries in

__two__or three dimensional Euclidean space

**.**

__Formally__

**,**EMF is classified as the external work expended per unit

**of**charge to produce

**an**electric potential difference across

__two__open-circuited terminals

**.**

__Formally__

**,**

**an**elliptic function is

**a**function meromorphic on for which there exist

__two__non-zero complex numbers

**and**with ( in other words

**,**not parallel ), such that

**and**for all

**.**

__Formally__

**,**

__two__variables are inversely proportional ( or varying inversely

**,**or in inverse variation

**,**or in inverse proportion or in reciprocal proportion ) if one

**of**the variables is directly proportional with the multiplicative inverse ( reciprocal )

**of**the other

**,**or equivalently if their product is

**a**constant

**.**

(

__Formally__speaking**,**this then satisfies the premises**of****an**axiom**of**well-founded induction**,**which asserts that these__two__conditions are sufficient for the proposition to hold for all x**.**__Formally__

**,**

**a**Lie superalgebra is

**a**( nonassociative ) Z

**<**

**sub**

**>**2

**</**

**sub**>- graded algebra

**,**or superalgebra

**,**over

**a**commutative ring ( typically R or

**C**) whose product

**,**called the Lie superbracket or supercommutator

**,**satisfies the

__two__conditions ( analogs

**of**the usual Lie algebra axioms

**,**with grading ):

__Formally__

**,**the poem can be understood as the combination

**of**

__two__sonnets

**,**though the spacing

**of**the stanzas is irregular

**.**

__Formally__

**,**

**a**vertex cover

**of**

**a**graph

**G**is

**a**set

**C**

**of**vertices such that each edge

**of**

**G**is incident to at least one vertex in

**C**

**.**The set

**C**is said to cover the edges

**of**

**G**

**.**The following figure shows examples

**of**vertex covers in

__two__graphs (

**and**the set

**C**is marked with red ).

__Formally__

**,**he says

**,**the piece

**consists**

**of**

__two__parts

**of**nearly equal length

**,**the end

**of**the first section being bars 24 – 28 ( p

**.**77 ).

__Formally__known as " Brooks Institute

**of**Photography ," Brooks Institute offers four majors

**,**

__two__certificate programs

**and**

__two__graduate programs

**.**

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