The Stacks project

Remark 63.5.6. Let $j : U \to X$ be an étale morphism of schemes. Then the trace map $\text{Tr} : j_!j^{-1}K \to K$ of Example 63.5.5 is equal to the counit for the adjunction between $j_!$ and $j^{-1}$. We already used the terminology “trace” for this counit in Étale Cohomology, Section 59.66.