My question is more about category theory and functors than about sheaves. We have a fixed topological space and we consider the categories of presheaves, sheaves and etale-sheaves over this topological space. In the second point of the theorem, is given a functor $\Phi$ which is, in some way, "the inverse" of the functor defined by the sheaf of section $\Gamma$. Here again, we see in brackets this functor is INJECTIVE on objects. In the third point we see the restriction of this functor $\Phi$ to the subcategory of sheaves is an isomorhpism of categories between sheaves and etale-sheaves.
I can't understand how a functor can be at the same time injective on objects and its restriction to a subcategory be an isomorphism. For sure this doesn't work for a classical injective function.
Moreover, the functor $\Phi$ gives the associated etale-sheaf of a presheaf. I'm not sure if it can be injective on objects at all. The associated etale-sheaf is built from the stalks, and we have more than one non-zero presheaf with zero stalks.
I hope I explained myself. Thank you in advance to those who can answer me.