Proof.
Via the identifications $\mathop{N\! L}\nolimits _{X/S} = \tau _{\geq -1}L_{X/S}$ (Lemma 92.20.4) and $H^0(L_{X/S}) = \Omega _{X/S}$ (Lemma 92.20.2) we have seen parts (2) and (3) in Deformation Theory, Lemmas 91.7.1 and 91.7.3.
Proof of (1). Roughly speaking, this follows from the discussion in Deformation Theory, Remark 91.7.9 by replacing the naive cotangent complex by the full cotangent complex. Here is a more detailed explanation. By Deformation Theory, Lemma 91.7.8 there exists an element
\[ \xi ' \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*\mathop{N\! L}\nolimits _{S/S'}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*L_{S/S'}, \mathcal{G}) \]
such that a solution exists if and only if this element is in the image of the map
\[ \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(NL_{X/S'}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(L_{X/S'}, \mathcal{G}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*L_{S/S'}, \mathcal{G}) \]
The distinguished triangle of Lemma 92.20.3 for $X \to S \to S'$ gives rise to a long exact sequence
\[ \ldots \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(L_{X/S'}, \mathcal{G}) \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*L_{S/S'}, \mathcal{G}) \to \mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_ X}(L_{X/S}, \mathcal{G}) \to \ldots \]
Hence taking $\xi $ the image of $\xi '$ works.
$\square$
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