Meaning of "espece de structure avec morphismes" in EGA Chapter 0 I was reading the beginning of EGA I, Chapter 0, on the section on sheaves, and came across this.

I cannot make heads of tails out of what Grothendieck is saying here.  An answer to a similar math overflow question https://mathoverflow.net/questions/11425/a-categorical-question gave a reference in Bourbaki's book on set theory.  Is it possible to understand this paragraph without reading that whole chapter on Bourbaki?  
Ideally, I would like to be able to paraphrase this section without having to go into a super technical discussion.
 A: Here is an English translation:
Suppose that $\mathbf{C}$ is a category defined by a "type of structure with morphisms" $\Sigma$.  Then the objects of $\mathbf{C}$ are the sets (of a given universe) equipped with structures of type $\Sigma$, and the morphisms of $\mathbf{C}$ are the morphisms of $\Sigma$.  Suppose as well that in $\mathbf{C}$, the kernel of a pair of morphisms $(u_1, u_2)$ exists and has as its underlying set the kernel of the pair of functions $(u_1,u_2)$.  Then condition (F) implies that, considered as a presheaf of sets, $\mathscr{F}$ is again a sheaf.  Moreover (in the notation of (F)), in order that a function $u: T \to \mathscr{F}(U)$ be a morphism of $\mathbf{C}$, it is necessary and sufficient, by virtue of (F), that each function $\rho_\alpha \circ u: T \to \mathscr{F}(U_\alpha)$ be a morphism, which means that the structure of type $\Sigma$ on $\mathscr{F}(U)$ is the initial structure for the family of morphisms $(\rho_\alpha)_\alpha$.  Conversely, suppose that a presheaf $\mathscr{F}$ on $X$ taking values in $\mathbf{C}$ is a sheaf of sets and satisfies the preceding condition.  Then it satisfies (F) and is thus a sheaf taking values in $\mathbf{C}$.
A: While one might perhaps object that this answer is cheap because very short, I believe it is very helpful: in https://arxiv.org/pdf/1509.08737v3.pdf you can find a detailed (yet only 4 page) description of the relation between modern category theory and the "espece de structure" of Bourbaki. I confirm what SpamIAm says: it is essentially just a concrete category, there is nothing deep in this.
