In this note, I want to record how to do the CY completion for a smooth and proper category $C$. Let $S: C \to C$ denote the Serre functor, which has the property that $$ Hom(x, y) = Hom(y, Sx)^\vee. $$

Given this endofunctor $S$, I can construct the free monad $$ T_S = id \oplus S \oplus S^2 \oplus \cdots $$ This acts on $C$, on object level, it sends $x$ to $x \oplus Sx \oplus S^2 x \cdots $ the gigantic thing. We have usual $id \to T_S$, and $T_s \otimes T_s \to T_s$. So it makes sense to consider monadic module over $T_S$. What is a module? It is an object $M$ in the category, together with the action morphism $T_S M \to M$.