In the case that $\mathcal{F}$ is cofibred in sets, i.e., corresponds to a functor $F : \mathcal{C}_\Lambda \to \textit{Sets}$ this follows from the description of (S2) for $F$ in Remark \ref{remark-compare-S1-S2-schlessinger} and the fact that $k[V] \cong k[\epsilon] \times_k \ldots \times_k k[\epsilon]$ with $\dim_k V$ factors. The case of functors is what we will use in the rest of this chapter. \medskip\noindent We prove the general case by induction on $\dim(V)$. If $\dim(V) = 1$, then $k[V] \cong k[\epsilon]$ and the result holds by assumption. If $\dim(V) > 1$ we write $V = V' \oplus k\epsilon$. Pick a diagram $$ \vcenter{ \xymatrix{ & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & k[V] \ar[d] \\ A \ar[r] & k } } $$ Choose a morphism $x_V \to x_{V'}$ lying over $k[V] \to k[V']$ and a morphism $x_V \to x_\epsilon$ lying over $k[V] \to k[\epsilon]$. Note that the morphism $x_V \to x_0$ factors as $x_V \to x_{V'} \to x_0$ and as $x_V \to x_\epsilon \to x_0$. By induction hypothesis we can find a diagram $$ \vcenter{ \xymatrix{ y' \ar[d] \ar[r] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\ A \ar[r] & k } } $$ This gives us a commutative diagram $$ \vcenter{ \xymatrix{ & x_\epsilon \ar[d] \\ y' \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & k[\epsilon] \ar[d] \\ A \times_k k[V'] \ar[r] & k } } $$ Hence by (S2) we get a commutative diagram $$ \vcenter{ \xymatrix{ y \ar[d] \ar[r] & x_\epsilon \ar[d] \\ y' \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ (A \times_k k[V']) \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\ A \times_k k[V'] \ar[r] & k } } $$ Note that $(A \times_k k[V']) \times_k k[\epsilon] = A \times_k k[V' \oplus k\epsilon] = A \times_k k[V]$. We claim that $y$ fits into the correct commutative diagram. To see this we let $y \to y_V$ be a morphism lying over $A \times_k k[V] \to k[V]$. We can factor the morphisms $y \to y' \to x_{V'}$ and $y \to x_\epsilon$ through the morphism $y \to y_V$ (by the axioms of categories cofibred in groupoids). Hence we see that both $y_V$ and $x_V$ fit into commutative diagrams $$ \vcenter{ \xymatrix{ y_V \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x_{V'} \ar[r]          & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ x_V \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x_{V'} \ar[r]                 & x_0 } } $$ and hence by the second part of (S2) there exists an isomorphism $y_V \to x_V$ compatible with $y_V \to x_{V'}$ and $x_V \to x_{V'}$ and in particular compatible with the maps to $x_0$. The composition $y \to y_V \to x_V$ then fits into the required commutative diagram $$ \vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\ A \ar[r] & k } } $$ In this way we see that the first part of $(S2)$ holds with $k[\epsilon]$ replaced by $k[V]$. \medskip\noindent To prove the second part suppose given two commutative diagrams $$ \vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r] \ar[d] & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\ A \ar[r] & k } } $$ We will use the morphisms $x_V \to x_{V'} \to x_0$ and $x_V \to x_\epsilon \to x_0$ introduced in the first paragraph of the proof. Choose morphisms $y \to y_{V'}$ and $y' \to y'_{V'}$ lying over $A \times_k k[V] \to A \times_k k[V']$. The axioms of a cofibred category imply we can find commutative diagrams $$ \vcenter{ \xymatrix{ y_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y'_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\ A \ar[r] & k } } $$ By induction hypothesis we obtain an isomorphism $b : y_{V'} \to y'_{V'}$ compatible with the morphisms $y_{V'} \to x$ and $y'_{V'} \to x$, in particular compatible with the morphisms to $x_0$. Then we have commutative diagrams $$ \vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_\epsilon \ar[d] \\ y'_{V'} \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r] \ar[d] & x_\epsilon \ar[d] \\ y'_{V'} \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\ A \ar[r] & k } } $$ where the morphism $y \to y'_{V'}$ is the composition $y \to y_{V'} \xrightarrow{b} y'_{V'}$ and where the morphisms $y \to x_\epsilon$ and $y' \to x_\epsilon$ are the compositions of the maps $y \to x_V$ and $y' \to x_V$ with the morphism $x_V \to x_\epsilon$. Then the second part of (S2) guarantees the existence of an isomorphism $y \to y'$ compatible with the maps to $y'_{V'}$, in particular compatible with the maps to $x$ (because $b$ was compatible with the maps to $x$).