  We prove a few results about the map $Spc(F)$ induced on tensor-triangular
spectra by a tensor-triangulated functor $F$. First, $F$ is conservative if and
only if $Spc(F)$ is surjective on closed points. Second, if $F$ detects
tensor-nilpotence of morphisms then $Spc(F)$ is surjective on the whole
spectrum. In fact, surjectivity of $Spc(F)$ is equivalent to $F$ detecting the
nilpotence of some class of morphisms, namely those morphisms which are
nilpotent on their cone.
