Definition 89.4.1. Let $\Lambda $ be a Noetherian ring and let $\Lambda \to k$ be a finite ring map where $k$ is a field. We define *$\widehat{\mathcal{C}}_\Lambda $* to be the category with

objects are pairs $(R, \varphi )$ where $R$ is a Noetherian complete local $\Lambda $-algebra and where $\varphi : R/\mathfrak m_ R \to k$ is a $\Lambda $-algebra isomorphism, and

morphisms $f : (S, \psi ) \to (R, \varphi )$ are local $\Lambda $-algebra homomorphisms such that $\varphi \circ (f \bmod \mathfrak m) = \psi $.

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