TITLE: Assumptions on the category C for sheafification of C-valued presheaves
QUESTION [11 upvotes]: For any category C and topological space X we have the notion of a C-valued presheaf on X.
What assumptions must be made about C in order that we have the notion of such a presheaf being a 'sheaf'? I understand the definition of the sheaf properties using an equalizer diagram which assumes C has products and a final object. Is this definition 'standard'?
Secondly, the definition of a sheafification of a presheaf in terms of the obvious universal property makes sense for any category C (for which the notion of sheaf makes sense). But what assumptions must be placed on C in order for such a sheafification to exist? For presheafs of sets I know the construction via the étale space of the presheaf (namely, the sheafification can be constructed as the sheaf of sections of the projection E->X of the etale space E onto X). This construction works in general right?

REPLY [2 votes]: Let PS(X,B) denote the category of presheaves on the category with values in category B. We denote by Sh((X,SX),B) the category of sheaves on the topos (CX,SX)(CX is category representing X and SX is pretopology). The assumption for category B is that 
B has filtered colimits which commutes with kernels of pairs of arrows. Let HX denote the endofunctor of PS(X,B) which assigns to every presheaf F: CX^op--->B of the presheaf HX(F) defined by 
HX(F)(N)=colim(Ker(F(M)-->-->F(M product M over N))),where N and M are objects of CX
where colimit is taken over Cov(N,t):(Cover of M, t is determined by SX mentioned above). We can easily check that Cov(N,t) is a filtered category. So the colimits we need is filtered colimits. 
We can also easily check that F(N)--->HX(F)(N) is a functor for each N belongs to ObCX
i.e. it defines a functor morphism F|--->HX(F)
This functor HX is called Heller functor. It is routine to check HX is left exact.
with following property 
1 Functor HX maps any presheaf to monopresheaf and maps any monopresheaf to sheaf. So sheafification functor is just HX compose HX.