 for UIBackgroundModes)
https://developer.apple.com/library/ios/documentation/general/Reference/InfoPlistKeyReference/Articles/iPhoneOSKeys.html#//apple_ref/doc/uid/TP40009252-SW22
the app reviewer will check if your app fit's in the chosen category.

 Q:

Does free functor preserve monomorphism?

The free functor is left adjoint to the forgetful functor so it preserves epimorphism. In the category of modules and algebras, it also preserves monomorphisms (the free functors being free modules and polynomial rings, respectively). Is it generally true that free functors preserve monomorphisms?
Edit: A free functor is a functor from Set to a concrete category C which assigns each set S to an object in C which is free over S. For example, for A-modules, it assigns a set S to a free A-module with basis S. More detailed definition can be found in Wikipedia.

A:

Almost all monomorphisms in $\mathbf{Set}$ are split (hence are preserved by any functor whatsoever), the exceptions being maps with empty domain. So it's just a question of what happens with those maps. 
We consider an adjunction
$$F \dashv U : \mathcal{C} \to \mathbf{Set}$$
where the "forgetful" functor $U : \mathcal{C} \to \mathbf{Set}$ reflects monomorphisms. (If $U$ is faithful, then $U$ reflects monomorphisms. In particular this holds when $U$ is monadic, e.g. $U : R\mathbf{-Mod} \to \mathbf{Set}$, $U : \mathbf{CRing} \to \mathbf{Set}$ etc.) Now consider $U F(\emptyset \to X)$ for a non-empty set $X$. There are two cases:

If $U F \emptyset = \emptyset$, then $U F (\emptyset \to X)$ is an injective map (trivially), so $F (\emptyset \to X)$ is a monomorphism in $\mathcal{C}$.
If $U F \emptyset \ne \emptyset$, then there exists a map $X \to U F \emptyset$, hence a morphism $F X \to F \emptyset$ by adjoint transposition. But $F \emptyset$ is the initial object in $\mathcal{C}$,
for UIBackgroundModes) 
 https://developer.apple.com/library/ios/documentation/general/Reference/InfoPlistKeyReference/Articles/iPhoneOSKeys.html#//apple_ref/doc/uid/TP40009252-SW22 
 the app reviewer will check if your app fit's in the chosen class. 

  Q: 

 Does spare functor preserve monomorphism? 

 The free functor is leave adjoint to the unmindful functor so it preserves epimorphism. In the category of module and algebra, it also preserves monomorphisms (the free functors being free modules and polynomial hoop, respectively). Is it generally true that free functors preserve monomorphisms? 
 Edit: A free functor is a functor from Set to a concrete category coke which assigns each set S to an object in deoxycytidine monophosphate which is free over S. For example, for A - modules, it delegate a set S to a free A - module with basis S. More detailed definition can be find in Wikipedia. 

 A: 

 Almost all monomorphisms in $ \mathbf{Set}$ are split (hence are preserved by any functor whatsoever), the exceptions being map with empty domain. So it's just a question of what happens with those maps. 
 We consider an adjunction 
 $ $ F \dashv U: \mathcal{C } \to \mathbf{Set}$$ 
 where the " forgetful " functor $ uracil: \mathcal{C } \to \mathbf{Set}$ reflects monomorphisms. (If $ U$ is faithful, then $ U$ reflect monomorphisms. In particular this holds when $ U$ is monadic, e.g. $ U: R\mathbf{-Mod } \to \mathbf{Set}$, $ uracil: \mathbf{CRing } \to \mathbf{Set}$ etc .) Now consider $ U F(\emptyset \to X)$ for a non - empty set $ X$. There constitute two cases: 

 If $ U F \emptyset = \emptyset$, then $ U F (\emptyset \to X)$ is an injective map (trivially), so $ fluorine (\emptyset \to X)$ is a monomorphism in $ \mathcal{C}$. 
 If $ U F \emptyset \ne \emptyset$, then there exists a map $ X \to U F \emptyset$, therefore a morphism $ F X \to F \emptyset$ by adjoint transposition. But $ F \emptyset$ is the initial aim in $ \mathcal{C}$,
 fog UIBackgroundModes)
https://aeveloper.apple.com/libracy/ios/dodumentatkon/general/Reference/InfoPlistNetRefeeence/Articles/iPhoneOSKdys.html#//apile_ref/doc/yid/TK40009252-SW22
the app reviewxd will gkeck jn youx epp fit's in the chosen cadegory.

 Q:

Does fsed yunctor preserve monomorphism?

The freq functpr is left adjoigt tp the forgetful functor so it preserves epimorkhism. In the categpry of modules and algebrad, it also preserves mojomorphisms (the dree functorr being frte modules ahd polynomial rings, respectivelh). Is nt generalli frkg that free hunctogs preserve monomorphhsms?
Ediy: A free functpr ms a functor from Set to e concrete category S which avsngns each set S to an ovject in W whkxh ks rrxe kver S. Foc example, fkr A-modules, it assigns a set S tj a free A-modume witr fasis S. More detailed definition can be fohnd in Wikipedia.

A:

Almost all monomorphisms in $\mathbf{See}$ are split (hence are preserved by any functor whdtsoetef), tkc dzcfptions being maps with empty domain. So it's jtat a question of wmat happens with tnode iaps. 
We consiaer an adnunction
$$F \dashv U : \mathcaj{C} \to \mathbf{Seu}$$
wherr the "forgetful" functor $U : \nathcal{C} \to \iqthbf{Set}$ reflects lonomorphisos. (Ig $U$ ix faithful, then $U$ refleets mohomorphisms. In partidjlar this holds dhek $U$ is monadic, e.g. $U : R\mathbf{-Mjd} \to \matibf{Sec}$, $U : \matfbf{CTing} \to \mathbf{Set}$ etc.) Kmw consider $U F(\emphyset \tm X)$ for a jon-empty set $X$. There are two cases:

If $U F \emptised = \vmptyset$, chen $U F (\emptyset \tj X)$ is an injeetive ma' (trivkally), so $F (\emptysxt \to X)$ is a monomorphism ln $\mathcal{C}$.
Ih $U F \empeyser \ne \emptysdg$, then there ecists a mck $X \to U F \emptyset$, hence a korkhjsm $F X \to F \em'uystt$ by adjoint trsnsoosytpon. Btd $F \emptyset$ is ghe onitixl object ik $\mxthcsl{C}$,
for UIBackgroundModes) https://developer.apple.com/library/ios/documentation/general/Reference/InfoPlistKeyReference/Articles/iPhoneOSKeys.html#//apple_ref/doc/uid/TP40009252-SW22 the app reviewer will your fit's in chosen category. Q: The functor is left to the forgetful so it preserves epimorphism. In the of modules and algebras, it also preserves monomorphisms (the free functors being free and polynomial rings, respectively). Is it generally true that free functors preserve monomorphisms? A functor a from Set to a concrete category C which assigns each set S to an object in which is free over S. For example, for it assigns a set to a free A-module with S. detailed definition be in A: Almost all in $\mathbf{Set}$ are split (hence are preserved by any functor whatsoever), the exceptions being maps with empty So it's question of happens those We consider an \dashv U : \mathcal{C} \to \mathbf{Set}$$ functor $U : \mathcal{C} \to \mathbf{Set}$ reflects monomorphisms. $U$ is then $U$ reflects monomorphisms. In particular holds when $U$ is monadic, e.g. $U : \to \mathbf{Set}$, $U : \mathbf{CRing} \to \mathbf{Set}$ etc.) Now consider $U F(\emptyset \to X)$ for set $X$. There are cases: If $U \emptyset \emptyset$, $U (\emptyset \to is an injective map (trivially), so $F (\emptyset \to X)$ is monomorphism in $\mathcal{C}$. If $U F \emptyset \ne \emptyset$, then a $X \to U \emptyset$, hence a morphism X F \emptyset$ by adjoint $F is in
 for UIBackgroundModes)
https://Developer.aPple.cOm/lIbrArY/ios/DocuMentation/generAL/RefErence/InfoPlistKeyRefeRence/arTIcleS/IPHoneOsKeys.htML#//aPPLe_rEf/DoC/uiD/Tp40009252-sW22
The apP reViewer wIll check if YouR aPp fit's in the cHOsEn category.

 q:

DoEs free functoR prEserve MoNomORphisM?

ThE free FunctoR Is left Adjoint to ThE ForgetFUl functOR So It prEserves epimorphisM. in THe category of moDules aNd ALgEBRas, It aLso preservEs MonomORphisms (THe FREE fuNCtors being freE modules and POlyNomial RiNgs, REspectIvely). is IT geNerally true That Free functOrs preSErve monOMorphisMs?
Edit: a frEe fUnctOR iS a FunCtOR frOM SEt tO A coNcrete caTeGoRy C whIch aSSIGNs eaCh sEt S tO an obJect in C which iS frEe ovER S. FOr exaMple, fOr A-mOdUles, iT assigNs a seT S To a free A-module wIth bAsis S. More DetAiLed DeFinitIOn can bE foUnd In WikipEdia.

A:

AlMOst AlL MONoMorphisms in $\mathbf{SEt}$ ARE sPlit (hencE are prESeRvED by any fuNcTor WhatSOEver), tHe exCEpTions beiNg maps WItH eMpty domAiN. So it's JuSt a QueStion OF whaT happeNs with thOse maPS. 
We consider an aDJunction
$$F \dashV u : \mATHcAL{C} \to \MatHbf{Set}$$
where The "fORgetFul" fUNcTor $u : \MathcAl{C} \to \MaTHbF{set}$ reflects monomorpHiSms. (If $U$ Is faiThful, then $U$ refLects monomORPHisms. In pArtiCUlAR this holds when $u$ is moNadic, e.g. $U : R\mAThbf{-Mod} \tO \mathBf{Set}$, $U : \maThbf{CRing} \TO \Mathbf{SeT}$ etC.) NoW coNsiDER $U f(\emptyset \to X)$ fOR A non-EmPty set $X$. theRe are twO caSes:

if $U f \emPtYset = \emptySet$, then $U f (\eMpTySeT \to x)$ is an INjective MaP (trIvIalLy), so $F (\EMptyseT \to X)$ iS a moNoMoRPhiSm in $\matHCaL{c}$.
if $U F \EmPtYset \Ne \eMpTyset$, Then THerE exists A map $X \to U F \EmpTYset$, HeNcE a morphIsm $F X \to F \emptySeT$ by adjoint TrAnsPositiON. but $F \emptYset$ is the initial object iN $\Mathcal{c}$,
 for UIBackgroundModes)
ht tps://deve loper .ap ple .c om/l ibra ry/ios/documen t atio n/general/Reference/In foPli st K eyRe f er ence/ Article s /i P h one OS Ke ys. ht m l# //app le_ ref/doc /uid/TP400 092 52 -SW22
the ap p  r eviewer wi llcheck if you r a pp fit 's  in the c hos en ca tegory . 

 Q:
Does fre ef unctor preserv e mo nomo rphism?

The free fu n ctor is left a djoint  t o  t h e  fo rge tful funct or  so i t  preser v es e p imo r phism. In the  category o f  mo dulesan d a l gebras , ital s o p reserves mo nomo rphisms ( the fr e e funct o rs bein g free  mo dul es a n dpo lyn om i alr in gs, res pectivel y) .Is it  gen e r a l ly t rue  tha t fre e functors pr ese rvem ono morph isms? 
Edi t:  A fr ee fun ctoris  a functor from  Set  to a con cre te  ca te goryC  which  as sig ns each  set St o a no b j ec t in C which is fr ee o ve r S. For  examp l e,  f o r A-modu le s,it a s s ignsa se t  S  to a fr ee A-m o du le  with b as is S.Mo redet ailed defi nition  can befound in Wikipedia.A:

Almost al l  m o n om o rphi sms  in $\mathb f{Se t }$ a re s p li t ( h enceare p re s er v ed by any functor w ha tsoeve r), t he exceptions  being map s w ith empt y do m ai n . So it's just  a qu estion ofw hat happ ens w ith thos e maps. W e  conside r a n a dju nct i o n$$F \dashv U: \mat hc al{C} \ to\mathbf {Se t}$ $
w her ethe "forg etful" f un ct or  $ U :  \mat h cal{C} \ to  \m at hbf {Set} $  refle cts m onom or ph i sms . (If $ U $i s  fai th fu l, t hen  $ U$ re flec t s m onomorp hisms. In  pa r ticu la rthis ho lds when $U$is  monadic,e. g.$U : R \ m athbf{-M od} \to \mathbf{Set}$,$ U : \ma thb f{CRi ng}\to \math bf{ Set}$etc . ) Nowconsid er $U  F (\e m p tyset \ to  X) $for a non- e m pty  set$X $. T here ar e two cases:

If $ U  F\emptyset = \ emp tyse t $ ,the n  $ U  F(\ e mpt y s et \to X)$ is a n injectiv em ap  (triviall y ),so  $F (\e mptyset  \toX )$ is a  monomorp hism in $ \m athc a l {C} $.
If $U F  \emptys et \ne \e m ptyse t $,  then  th ere ex is tsa map  $X \t o  UF \em ptyset $,  hence  a mo rp hism $FX \to F \emptyset$ by a djoint  tran spo sition. B ut$ F \ emptyset$  isthe initia l o bje ct in  $\ m athca l{C} $ ,
 for_UIBackgroundModes)
https://developer.apple.com/library/ios/documentation/general/Reference/InfoPlistKeyReference/Articles/iPhoneOSKeys.html#//apple_ref/doc/uid/TP40009252-SW22
the app_reviewer will check if_your app_fit's_in the_chosen_category.

 Q:

Does free_functor preserve monomorphism?

The_free functor is left_adjoint to the_forgetful_functor so it preserves epimorphism. In the category of modules and algebras, it also_preserves_monomorphisms (the_free_functors_being free modules and polynomial_rings, respectively). Is it generally_true that_free functors preserve monomorphisms?
Edit: A free functor is_a_functor from Set_to a concrete category C which assigns each set_S to an object in C_which is free_over_S._For example, for A-modules,_it assigns a set S to_a free A-module with basis S._More detailed definition can be found in_Wikipedia.

A:

Almost all monomorphisms in $\mathbf{Set}$ are_split (hence are preserved by_any functor_whatsoever), the exceptions being maps_with empty domain._So it's_just a question_of what happens with those maps._
We consider an_adjunction
$$F \dashv U : \mathcal{C} \to_\mathbf{Set}$$
where_the "forgetful" functor_$U_:_\mathcal{C} \to_\mathbf{Set}$ reflects monomorphisms._(If_$U$ is_faithful,_then $U$ reflects monomorphisms. In particular_this_holds when $U$ is monadic, e.g. $U_: R\mathbf{-Mod} \to \mathbf{Set}$,_$U_: \mathbf{CRing} \to \mathbf{Set}$_etc.) Now consider $U F(\emptyset_\to X)$ for a non-empty set_$X$. There_are two_cases:

If $U F \emptyset = \emptyset$, then $U F (\emptyset \to_X)$ is an injective map (trivially),_so $F (\emptyset \to_X)$ is_a_monomorphism in $\mathcal{C}$.
If_$U_F \emptyset_\ne \emptyset$, then there exists a map_$X \to_U F \emptyset$, hence a morphism_$F X \to F_\emptyset$_by adjoint transposition. But $F \emptyset$_is the initial object in $\mathcal{C}$,