Lemma 91.11.4. Let $(f, f')$ be a morphism of first order thickenings as in Situation 91.9.1. Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}'$-modules and set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}$-linear map. Assume that $\mathcal{G}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$ and that $(f, f')$ is a strict morphism of thickenings. The set of lifts of $\varphi $ to an $\mathcal{O}'$-linear map $\varphi ' : \mathcal{F}' \to \mathcal{G}'$ is, if nonempty, a principal homogeneous space under

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G} \otimes _\mathcal {O} f^*\mathcal{J}) \]

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