Lemma 47.17.1. In Situation 47.16.1 let $Y$ be an object of $\textit{FTS}_ S$ and let $j : X \to Y$ be an open immersion. Then there is a canonical isomorphism $j^! = j^*$ of functors.

Proof. In this case we may choose $\overline{X} = Y$ as our compactification. Then the right adjoint of Lemma 47.3.1 for $\text{id} : Y \to Y$ is the identity functor and hence $j^! = j^*$ by definition. $\square$

Comment #4632 by Noah Olander on

I think it's useful to point out either here or later that by the end of the chapter you know that $f^! = f^*$ for $f$ etale in $FTS_S$.

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