In SGA 1, Expose 5, there is a proposition (3.5), which states that for any two objects $X$ and $X'$ in $\textbf{Et}/S$, all $S$ morphisms between them factor into a surjective etale morphism $X\to X''$ and a canonical immersion $X''\to X'$. If I am understanding this proof correctly, this is by the fact that to be in $\textbf{Et}/S$, any morphism from an object in the category to $S$ is finite etale, and therefore both open and closed, and so any morphism between objects in the category is both open and closed, so our image will be open and closed, and thus a finite union of connected components of $X'$.

There is a corollary following this proposition, rephrasing it in terms of an effective epimorphism followed by a monomorphism. What are the advantages of this? The corollary follows by the fact that the first map is faithfully flat, and thus a strict epimorphism, and a similar result for the monomorphism. Perhaps this is just my lack of familiarity with category theory, but would every strict epimorphism be a faithfully flat morphism in this category? Would the reason to introduce these terms just be to define Galois categories later, and should we always just think about strict epimorphisms/monomorphisms in the sense of Proposition 3.5?

(Also, is there a typo in the statement, switching the roles of $X'$ and $X''$?)