In the setting described in Bernstein, Beilinson, and Deligne, associated to a scheme $X$, a closed subscheme $i: Z \to X$ and its open complement $j: U \to X$ we have six functors between the corresponding derived categories of étale sheaves $i^*, i_*, i^!, j_!, j^*, j_*$ (they are all derived but I will ommit the L's and R's).

My question is: if $X$ is a variety over a field $k$ with structural morphism $f: X \to k$ and $F$ is an object of $D^b(k)$, then is the canonical morphism $f^*F \to i_*i^*f^*F \oplus j_*j^*f^*F$ ever a monomorphism in the triangulated category $D^b(X)$?