Coordinate-shift shadow containment ($\partial C_{ij}\mathcal{F}\subseteq C_{ij}\partial \mathcal{F}$)

For a uniform set family, the lower shadow of its shifted set family is contained in the shifted lower shadow. Check the witness in the four patterns of membership in coordinates $i,j$. A surviving shadow member containing only $j$ must have both old shadow partners; a member containing only $i$ needs either old partner. Taking complements exchanges $C_{ij}$ with $C_{ji}$ and gives the same containment for the upper shadow. Thus coordinate shifts of a set family cannot increase either shadow or their disjoint union, the external vertex boundary of a uniform set family.