Proof.
It is immediate that (2) implies (1). Assume \{ Y_ i \to Y\} is as in (1) and let W \to Y be as in (2). Then \{ Y_ i \times _ Y W \to W\} _{i \in I} is an étale covering, which we may refine by an étale covering \{ W_ j \to W\} _{j = 1, \ldots , m} with W_ j affine (Topologies, Lemma 34.4.4). Thus to finish the proof it suffices to show the following three algebraic statements:
if R \to A \to B are ring maps with A \to B étale and A glueable as an R-module, then B is glueable as an R-module,
finite products of glueable R-modules are glueable,
if R \to A \to B are ring maps with A \to B faithfully étale and B glueable as an R-module, then A is glueable as an R-module.
Namely, the first of these will imply that \Gamma (W_ j, \mathcal{O}_{W_ j}) is a glueable R-module, the second will imply that \prod \Gamma (W_ j, \mathcal{O}_{W_ j}) is a glueable R-module, and the third will imply that \Gamma (W, \mathcal{O}_ W) is a glueable R-module.
Consider an étale R-algebra homomorphism A \to B. Set A' = A \otimes _ R R' and B' = B \otimes _ R R' = A' \otimes _ A B. Statements (1) and (3) then follow from the following facts: (a) A, resp. B is glueable if and only if the sequence
0 \to A \to A_ f \oplus A' \to A'_ f \to 0, \quad \text{resp.}\quad 0 \to B \to B_ f \oplus B' \to B'_ f \to 0,
is exact, (b) the second sequence is equal to the functor - \otimes _ A B applied to the first and (c) (faithful) flatness of A \to B. We omit the proof of (2).
\square
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