what is meant by a pro-object "normalisé de la facon habituelle" in SGA 1, Expose V?

Question

My page numbers will refer to the nice typed-up version of SGA found at: http://arxiv.org/abs/math/0206203

His first mention of "pro-objet" (pro-object) is on page 99. In prop 5.3 (p105), he talks about the pro-objet associated to a fundamental functor, "normalisé de la facon habituelle" (normalized in the usual way). He uses the same phrase in prop 5.5 (p106), and in prop 5.6 (p106), he talks about an arbitrary pro objet, "mis sous forme normale" (put in normal form). What does he mean by this normal form?

Answer 1

The answer is on page 99 and in Technique de descente et théorèmes d’existence en Géométrie Algébrique, II (Section 3): the upshot is that the functor $F$ is not only pro-representable, it is strictly pro-representable in the sense of loc. cit. section 2 (especially the end of the section), a definition which is also recalled on page 99 of (the arxiv version of) SGAI. When a functor is strictly pro-representable, it is represented by a unique pro-object (up to unique isomorphism) satisfying the supplementary conditions that the transition morphisms be epimorphisms and that epimorphisms $P_i\rightarrow P'$ be equivalent to epimorphisms $P_i\rightarrow P_j$ for suitable $j$.

As the fibre functor is strictly pro-representable, it is represented by a unique such object (up to unique isomorphism) and this is that choice of $P$ which is the usual normalization alluded to in the text (without the uniqueness provided by this normalization, Proposition 5.6 would be false, for instance).