Let $S,S'$ be schemes, let $\pi : S' \to S$ be a morphism which is faithfully flat and locally of finite presentation, set $S'' := S' \times_{S} S'$ and $S''' := S' \times_{S} S' \times_{S} S'$ with projections $p_{1},p_{2} : S'' \to S'$ and $p_{12},p_{13},p_{23} : S''' \to S''$. Let $\mathcal{E}'$ be a vector bundle on $S'$ such that there is an isomorphism $\varphi : p_{1}^{\ast}\mathcal{E}' \to p_{2}^{\ast}\mathcal{E}'$ of $\mathcal{O}_{S''}$-modules. What's an example of such $S,S',\pi,\mathcal{E}'$ such that $\mathcal{E}'$ does not descend to $S$, i.e. there does not exist a vector bundle $\mathcal{E}$ on $S$ and an $\mathcal{O}_{S'}$-module isomorphism $\pi^{\ast}\mathcal{E} \simeq \mathcal{E}'$?

*Thoughts*:

- Such $\mathcal{E}$ exists if and only if there exists some $\psi \in \operatorname{Aut}_{\mathcal{O}_{S''}}(p_{1}^{\ast}\mathcal{E}')$ such that the cocycle condition $p_{23}^{\ast}(\varphi \circ \psi) \circ p_{12}^{\ast}(\varphi \circ \psi) = p_{13}^{\ast}(\varphi \circ \psi)$ is satisfied.
- If $\pi$ admits a section $\sigma : S \to S'$, then we can take $\mathcal{E} = \sigma^{\ast}\mathcal{E}'$ and pull back $\varphi$ via an induced section $S' \to S''$.
- If $S = \operatorname{Spec} k$ and $S'$ is a smooth projective geometrically integral $k$-scheme and $\mathcal{E}'$ is a line bundle, then $\operatorname{Pic}(S') \times \operatorname{Pic}(S') \to \operatorname{Pic}(S'')$ is injective so the existence of $\varphi$ implies that $\mathcal{E}' \simeq \mathcal{O}_{S'}$.