Reading the book of Goess-Jardine or of Gabriel-Zisman on the simplicial homotopy there are coker presentation of the boundary $\partial\Delta^n $ of the elementary simplex $\Delta^n $ or the horn $\Lambda^n_k $ (see GOerss-Jardine "Simplicial Homotopy Theory" pag.9) .

Now the image of this presentation coker diagram by the topological realization functor $T: S \to Top$ (where $S$ category of functors from $\Delta^{ op }$ to $Set$, and $Top$ category of topological spaces) is of course still a coker diagram (the functor $T$ preserves colimits being a left adjoint) , a separate verification that this latter is a coequalizers (of topological spaces) is evident (or at most geometrically intuitive). Of course is a no hard problem give a demonstration that also the original presentation (by simplicial spaces) is a coequalizer.

Anyway this little question has suggested me the follow question:

Let $\iota: \coprod \Delta \subset S$ the subcategory by object the finite coproducts of representales , and by morphisms constructed from morphisms between representable and coprojections. Considering the restriction functor $T\circ \iota: \coprod \Delta \to Top $, the question is:

This functors reflex cokernels?