Remark 89.17.6. The action of Lemma 89.17.5 is functorial. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of deformation categories. Let $A' \to A$ be a surjective ring map whose kernel $I$ is annihilated by $\mathfrak m_{A'}$. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$. In this situation $\varphi $ induces the vertical arrows in the following commutative diagram
\[ \xymatrix{ \text{Lift}(x, A') \times (T\mathcal{F} \otimes _ k I) \ar[d]_{(\varphi , d\varphi \otimes \text{id}_ I)} \ar[r] & \text{Lift}(x, A') \ar[d]^\varphi \\ \text{Lift}(\varphi (x), A') \times (T\mathcal{G} \otimes _ k I) \ar[r] & \text{Lift}(\varphi (x), A') } \]
The commutativity follows as each of the maps (89.17.5.2), (89.17.5.1), and (89.17.5.3) of the proof of Lemma 89.17.5 gives rise to a similar commutative diagram.
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