Dear Vaino,

let F be a fixed field. Consider the category of field extensions L over F. Objects are fields L containing F as a subfield, morphisms are field inclusions L->K over F (identical on F). Then consider the "dual category" with objects L*=Hom(L,F) (dual space of F-linear mappings). Any inclusion L->K gives rise to the corresponding surjection K*->L*. Something like this.

Maybe it looks a bit artificial but I hope it can help.

Best regards, Yuri 

Since surjectivity (or epi-ness) is preserved under composition you can just take any category and take the subcategory of the category that has epimorphisms as morphisms.  This is a party trick. However, you can also consider the dual subcategory of monomorphisms, and since every morphism can be factored as an epi followed by a mono studying these two sub categories gives you a handle on the whole category. You also  often have a preferred subcategory of isomorphisms lying around and you may ask good behaviour with respect to those. Turns out this is the basic idea of model categories which is a very nice abstract of homological algebra phrased in the geometric language of homotopy theory (it is abstract homotopy theory actually). The book by Mark Hovey was the canonical reference last time I checked. 

@Vaino: P.S. Which Normal do you mean, there are too many: normal operators, normal varieties, normal spaces, ...

It isn't quite clear what means `where'. Does it mean `where from' or `where to' or `between'?. If `where from' then take maximal structure each homo from which is regular mono. If it means `where to' then take minimal structure. If `both' then take a structure which is both maximal and minimal. In any case, the structure of (linearly ordered) continuous field is what you need: every homo from any continuous field to any continuous field is iso.

A very simple example is the complex n x n matrices M_n. Every homomorphism of M_n into itself is an isomorphism since M_n is simple, and it is surjective because the image has the same dimension as M_n. The argument can be extended to other simple finite dimensional algebras.

Dear Vaino,

you modified your question but after that, it does not become more clear. Do you understand what you want? It seems that you think that for each structure species there exists canonically associated notion of homomorphism. This is not the case! For any class of objects, there exist many ways to associate with them homomorphisms. You can take as homos all maps, all bijective maps, all injective maps, all surjective maps, etc. So, let me repeat it, what do you want?    

One can consider all sets and only the surjective functions as morphisms.

@mihai that is the simplest example of the construction I started with.

Let me put it this way: I want an algebraic structure in which  (if it makes sense to talk about kernels )  kernels are preserved under  arbitrary homomorphisms. @Aslanbek

If you want to preserve kernels, @Erling Stormer gave an idea here:

Simple structures like simple algebras (in particular Matrix algebras), von Neumann factors,  simple groups, simple Lie algebra's etc. Either everything is the kernel or the kernel is trivial. And there are obviously non trivial examples.  

Thank you I will study that @ Rogier

 But in a case of simple groups kernels are not preserved at all. Ofcouse it is true if one consider a homomorphism from a simple group into it self  @Rogier