Remark 61.5.6. 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 61.5.5 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 58.66.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).