The Stacks project

Example 90.16.10. A prorepresentable functor $F$ is a deformation functor. Namely, suppose $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$ and $F(A) = \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(R, A)$. There is a unique morphism $R \to k$, so $F(k)$ is a one element set. Since

\[ \mathop{\mathrm{Hom}}\nolimits _\Lambda (R, A_1 \times _ A A_2) = \mathop{\mathrm{Hom}}\nolimits _\Lambda (R, A_1) \times _{\mathop{\mathrm{Hom}}\nolimits _\Lambda (R, A)} \mathop{\mathrm{Hom}}\nolimits _\Lambda (R, A_2) \]

the same is true for maps in $\widehat{\mathcal{C}}_\Lambda $ and we see that $F$ has (RS).

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