Suppose we have a scheme $X$ and a closed subscheme $Z$, with complement $U$. Then, for any étale sheaf $F$ on $X$, we get a long exact sequence in cohomology
$\cdots H^i(X,F) \to H^i(U,F) \to H^{i+1}_Z(X,F) \to H^{i+1}(X,F) \to \cdots$
as described for example by Theorem 9.4 in Milne's Lecture Notes on Étale Cohomology. (Do we maybe need $Z$ to be reduced?) Milne also says that the sequence is functorial in the pair $(X,X \setminus Z)$. My question is:
What does this mean? In other words, what notion of morphism of pairs makes this statement true?
[Edit: the claimed counterexample below is not one, as pointed out by Ulrich.]
The reason I think clarification is needed is the following example: take $X$ to be the affine line over an algebraically closed field of characteristic zero and $Z$ to be the origin. Then there is a functorial isomorphism $H^2_Z(X, \mu_n) \cong H^0(Z,\mathbb{Z}/n)$ given by the purity theorem, and the exact sequence gives an isomorphism $H^1(U,\mu_n) \to H^0(Z,\mathbb{Z}/n) = \mathbb{Z}/n$. Now let $f \colon X \to X$ be the map given by $x \mapsto x^n$. This seems to induce the zero map on $H^1(U,\mu_n)$ but the identity map on $H^0(Z,\mathbb{Z}/n)$, and so the sequence doesn't seem to be functorial in this instance.
