. However the converse need not hold; the smaller category can (and often will) have more epimorphisms.

As for most concepts in category theory, epimorphisms are preserved under equivalences of categories: given an equivalence F : C → D, a morphism f is an epimorphism in the category C if and only if F(f) is an epimorphism in D. A duality between two categories turns epimorphisms into monomorphisms, and vice versa.

The definition of epimorphism may be reformulated to state that f : X → Y is an epimorphism if and only if the induced maps

are injective for every choice of Z. This in turn is equivalent to the induced natural transformation

being a monomorphism in the functor category SetC.

Every coequalizer is an epimorphism, a consequence of the uniqueness requirement in the definition of coequalizers. It follows in particular that every cokernel is an epimorphism. The converse, namely that every epimorphism be a coequalizer, is not true in all categories.

In many categories it is possible to write every morphism as the composition of an epimorphism followed by a monomorphism. For instance, given a group homomorphism f : G → H, we can define the group K = im(f) and then write f as the composition of the surjective homomorphism G → K that is defined like f, followed by the injective homomorphism K → H that sends each element to itself. Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in all abelian categories and also in all the concrete categories mentioned above in  (though not in all concrete categories).

Related concepts 
Among other useful concepts are regular epimorphism, extremal epimorphism, immediate epimorphism, strong epimorphism, and split epimorphism. 

 An epimorphism is said to be regular if it is a coequalizer of some pair of parallel morphisms. 
 An epimorphism  is said to be extremal if in each representation, where  is a monomorphism, the morphism  is automatically an isomorphism. 
 An epimorphism  is said to be immediate if in each representation, where  is a monomorphism and  is an epimorphism, the morphism  is automatically an isomorphism.
  An epimorphism  is said to be strong if for any monomorphism  and any morphisms  and  such
. However the converse need not hold; the smaller category can (and often will) get more epimorphisms. 

 As for most concept in category theory, epimorphisms are save under equivalence of categories: given an comparison fluorine: C → D, a morphism f is an epimorphism in the category C if and only if F(f) is an epimorphism in D. A duality between two categories turn epimorphisms into monomorphisms, and vice versa. 

 The definition of epimorphism may be redevelop to state that f: X → Y is an epimorphism if and only if the induced maps 

 are injective for every choice of Z. This in turn is equivalent to the induce natural transformation 

 being a monomorphism in the functor category SetC. 

 Every coequalizer is an epimorphism, a consequence of the uniqueness prerequisite in the definition of coequalizers. It follows in particular that every cokernel is an epimorphism. The converse, namely that every epimorphism be a coequalizer, is not true in all categories. 

 In many class it is possible to write every morphism as the composition of an epimorphism followed by a monomorphism. For instance, give a group homomorphism f: G → H, we can define the group K = im(f) and then write f as the composition of the surjective homomorphism G → K that is defined like f, follow by the injective homomorphism K → H that sends each element to itself. Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in all abelian categories and also in all the concrete categories mentioned above in   (though not in all concrete categories). 

 Related concept 
 Among early useful concepts are regular epimorphism, extremal epimorphism, contiguous epimorphism, impregnable epimorphism, and split epimorphism. 

  An epimorphism is said to be regular if it is a coequalizer of some pair of parallel morphisms. 
  An epimorphism   is said to be extremal if in each representation, where   is a monomorphism, the morphism   is automatically an isomorphism. 
  An epimorphism   is say to be immediate if in each representation, where   is a monomorphism and   is an epimorphism, the morphism   is automatically an isomorphism. 
   An epimorphism   is said to be strong if for any monomorphism   and any morphisms   and   such
. Hoaever the converse need kot hold; the smaller cavegory dan (and uften will) have more epimorpiismw.

As fir most concepts in cagegory thvory, epimirphmsms are preservxs under equivzpenczs of categories: given an aquivalence F : C → B, a morphism f is an epimorphism in ehe catrglry C if and ogly pf F(f) js an epimorphism in D. A duality bstween uwo categories turms epimorphisms into monomlrphlsms, and vice versw.

The definijjon if epimorphirm may be gzformulated to state that f : X → Y is an epkmorpkism if and obly hf the indured mais

are injectiyv for eeery chpice of Z. This in tuen is equivalent to tie induced natural ttansformathou

being a monomorphism ib the funwtor xatdgody SstC.

Evegy roequalizer is an epimirphism, a consequenve if the uniquehess rqqtirement in the definition of coequaliztrs. If follows in particular that every cokernel ls an epiiorphism. The converse, namely that every epimorphivm be x ciezjqllzer, is not true in all categories.

In many catqfotivs it is possible to write efegy iorphism as tfe com'ksjtion of an epimorohism fjllowwd by a mjnomprphism. For instance, given q group homoiirphism f : G → H, we can define the groul K = im(f) and then write f aa the compodition of ghe surjective humokosphism G → K that is defineq like f, holloced by tfe imjectide homomorohism K → H that sends eafh elgment do itself. Duch a factorization of an arbivcary morphism itto an epimjrphixm followed br a monomorphixm can ye carfied out ih all auelian categjries and alsm in all the roncrete satetoriws mentkuned above in  (though not in all xoncrete categoriex).

Reuzted concepts 
Anont other useful vonzepes ace redglar epimorpvism, exgtemal epimorkkisn, immrdiate epimorphism, sdronf epimorphism, and xpjit epimirphism. 

 An epimorphixm is said to be rtgular if it ix a coequalizer of some pair of pzrallel mlrpmisms. 
 An epiiorpmism  is said co be extremal if in each representation, where  is a monomirphism, the morphisk  is automavicallr an isommrphism. 
 An epimorphusm  is said to bt immediate if in each depresantatlon, where  is a monomorphism and  is an epimorphism, the morphism  is autonatically an isomodphixm.
  At zpikorphifm  ms said to be stromg if for any monomorphism  and any mor[hnsms  and  such
. However the converse need not hold; category (and often have more epimorphisms. category epimorphisms are preserved equivalences of categories: an equivalence F : C → a morphism f is an epimorphism in the category C if and only F(f) is an epimorphism in D. A duality between two categories turns epimorphisms monomorphisms, vice The of epimorphism may be reformulated to state that f : X → Y is an epimorphism and only if the induced maps are injective every choice of Z. in turn is equivalent to induced transformation being monomorphism the category SetC. Every is an epimorphism, a consequence of the uniqueness requirement in the definition of coequalizers. It follows in that every an epimorphism. converse, that epimorphism be a not true in all categories. In is possible to write every morphism as the of an followed by a monomorphism. For instance, a group homomorphism f : G → H, can define the group K = im(f) and then write f as the composition of homomorphism G → K is defined like followed the homomorphism → H sends each element to itself. Such a factorization of an arbitrary into an epimorphism followed by a monomorphism can be carried all categories and also all the concrete categories above (though not in all Related Among are epimorphism, epimorphism, immediate epimorphism, strong and split epimorphism. An epimorphism said to be regular of some pair of parallel morphisms. An epimorphism said to be extremal if in each where is a monomorphism, the morphism is automatically an isomorphism. An epimorphism said to if in each representation, where is a monomorphism is an epimorphism, the is automatically an isomorphism. An epimorphism is said to strong for any and any morphisms such
. However the converse need not Hold; the smaLler cAteGorY cAn (anD oftEn will) have more EPimoRphisms.

As for most concepTs in cAtEGory THeOry, epImorphiSMs ARE prEsErVed UnDEr EquivAleNces of cAtegories: gIveN aN equivalence f : c → D, A morphism f Is aN epimorphism In tHe cateGoRy C IF and oNly If F(f) iS an epiMOrphisM in D. A dualItY BetweeN Two cateGORiEs tuRns epimorphisms inTO mONomorphisms, and Vice veRsA.

thE DEfiNitIon of epimoRpHism mAY be refoRMuLATEd tO State that f : X → Y iS an epimorphISm iF and onLy If tHE inducEd mapS

aRE inJective for eVery Choice of Z. this in TUrn is eqUIvalent To the iNduCed NatuRAl TrAnsFoRMatIOn

BeiNG a mOnomorphIsM iN the fUnctOR CATegoRy SEtC.

EVery cOequalizer is aN epImorPHisM, a conSequeNce oF tHe uniQuenesS requIrEment in the definItioN of coequaLizErS. It FoLlows IN partiCulAr tHat everY cokernEL is An EPIMoRphism. The converse, nAmELY tHat every EpimorPHiSm BE a coequaLiZer, Is noT TRue in All cATeGories.

In Many caTEgOrIes it is PoSsible To WriTe eVery mORphiSm as thE composiTion oF An epimorphism fOLlowed by a monoMOrPHIsM. for iNstAnce, given a gRoup HOmomOrphISm F : G → H, WE can dEfine ThE GrOUp K = im(f) and then write f As The comPositIon of the surjeCtive homomORPHism G → K thAt is DEfINed like f, followEd by tHe injectivE HomomorpHism K → h that senDs each eleMENt to itseLf. SUch A faCtoRIZaTion of an arbitRARy moRpHism intO an EpimorpHisM foLloWed By A monomorpHism can bE cArRiEd Out In all ABelian caTeGorIeS anD also IN all thE concRete CaTeGOriEs mentiONeD ABove In  (ThOugh Not In All coNcreTE caTegorieS).

Related cOncEPts 
AMoNg Other usEful concepts aRe Regular epiMoRphIsm, extREMal epimoRphism, immediate epimorphISm, stronG epImorpHism, And split ePimOrphisM. 

 An EPimorpHism is Said tO bE reGULar if IT Is A coEqUalizer of sOME paIr of pArAlleL morphiSms. 
 An epimorphism  is SAid To be extremal iF in Each REPrEseNTaTIon, WhERe  iS A Monomorphism, the Morphism  is AuTOmAtically an ISomOrPhism. 
 An EpimorpHism  iS Said to bE immediatE if in each RePresENTatIon, where  is A monomorPhism and  iS An epiMOrPhism, The MorphiSm  Is aUtomaTicallY An iSomorPhism.
  AN ePimorpHism  iS sAid to be sTrong if for any monomorphiSm  and aNy morPhiSms  and  sucH
. However the converse nee d not hold ; the  sm all er  cat egor y can (and oft e n wi ll) have more epimorph isms. 
A s fo r  m ost c oncepts in c ate go ry  th eo r y,  epim orp hisms a re preserv edun der equivale n ce s of categ ori es: given an  eq uivale nc e F : C →  D,  a mo rphism f is a n epimorp hi s m in t h e categ o r yC if  and only if F(f) is an epimorphism  in D.  A du a l ity  be tween twoca tegor i es turn s  e p i m orp h isms into mon omorphisms, and  viceve rsa . 

Thedefin it i onof epimorph ismmay be re formul a ted tos tate th at f :  X→ Y  isa nep imo rp h ism if  an d  on ly if th ein duced  map s a re i nje ctiv e for  every choice  of  Z.T his  in t urn i s eq ui valen t to t he in du ced natural tra nsfo rmation
bei ng  amo nomor p hism i n t hefunctor  catego r y S et C . 
E very coequalizer i sa n  e pimorphi sm, ac on se q uence of  t heuniq u e nessrequ i re ment inthe de f in it ion ofco equali ze rs.  It  foll o ws i n part icular t hat e v ery cokernel i s  an epimorphi s m. T he conv ers e, namely t hate very  epi m or phi s m bea coe qu a li z er, is not true inal l cate gorie s.

In many c ategoriesi t is possi blet ow rite every mor phism  as the co m position  of a n epimor phism fol l o wed by a  mo nom orp his m .  F or instance,g i venagroup h omo morphis m f  :G →  H,  w e can def ine thegr ou pK= i m(f)a nd thenwr ite  f  as  thec omposi tionof t he  s u rje ctive h o mo m o rphi sm  G  → K  th at  is d efin e d l ike f,followedbyt he i nj ec tive ho momorphism K→H that sen ds  ea ch ele m e nt to it self. Such a factorizat i on of a n a rbitr arymorphismint o an e pim o rphism  follo wed b ya m o n omorp h i sm  ca nbe carried o utin al label ian cat egories and also i n  al l the concret e c ateg o r ie s m e nt i one da bov e in  (though not  in all co nc r et e categori e s). 
Related  concep ts 
A m ong oth er useful  concepts  a re r e g ula r epimorph ism, ext remal epi m orphi s m,  imme dia te epi mo rph ism,strong epi morph ism, a nd  split  epim or phism. 
 An epimorphism is sai d to b e reg ula r if it i s a coe qualizerof s ome pair o f p ara llelmor p hisms . 
A nepi m orphi sm i s said to be  ex t r em al if in ea c h rep resen tat i on, wh ere is a monomorphis m , the morphism   is a uto mat i call yan isomorphism .  A n epimorph is m  is saidto be im me d iateif ineach r epresen t a ti o n, whe re isa monomor phi sm and  is  a ne pimorp hism ,the mo rphism  isa u tomatically an i somor p h ism. An  epim or phism i s sa id to be s trong if fo r anymono morph ism  an dany mo rph is ms  and  s u ch
. However_the converse_need not hold; the_smaller category_can_(and often_will)_have more epimorphisms.

As_for most concepts_in category theory, epimorphisms_are preserved under_equivalences_of categories: given an equivalence F : C → D, a morphism f is_an_epimorphism in_the_category_C if and only if_F(f) is an epimorphism in_D. A_duality between two categories turns epimorphisms into monomorphisms,_and_vice versa.

The definition_of epimorphism may be reformulated to state that f_: X → Y is an_epimorphism if and_only_if_the induced maps

are injective_for every choice of Z. This_in turn is equivalent to the_induced natural transformation

being a monomorphism in the_functor category SetC.

Every coequalizer is an_epimorphism, a consequence of the_uniqueness requirement_in the definition of coequalizers._It follows in_particular that_every cokernel is_an epimorphism. The converse, namely that_every epimorphism be_a coequalizer, is not true in_all_categories.

In many categories_it_is_possible to_write every morphism_as_the composition_of_an epimorphism followed by a monomorphism._For_instance, given a group homomorphism f :_G → H, we_can_define the group K_= im(f) and then write_f as the composition of the_surjective homomorphism_G →_K that is defined like f, followed by the injective homomorphism_K → H that sends each_element to itself. Such_a factorization_of_an arbitrary morphism_into_an epimorphism_followed by a monomorphism can be carried_out in_all abelian categories and also in_all the concrete categories_mentioned_above in  (though not in_all concrete categories).

Related concepts 
Among other_useful concepts are regular epimorphism,_extremal_epimorphism,_immediate epimorphism, strong epimorphism, and_split epimorphism. 

 An epimorphism is_said to be_regular if it is a coequalizer of_some_pair of parallel morphisms. 
 An_epimorphism_ is said to be extremal_if_in_each representation, where  is_a monomorphism, the morphism  is_automatically an isomorphism. 
 An epimorphism  is said_to be immediate_if in each representation, where__is_a monomorphism and  is an epimorphism, the morphism _is automatically_an isomorphism.
 _An epimorphism  is said to be strong if for_any monomorphism  and any morphisms _and  such