  In this note we generalize a result from a recent paper of Hajac, Reznikoff
and Tobolski (2020). In that paper they give conditions they call admissibility
on a pushout diagram in the category of directed graphs implying that the
$C^*$-algebras of the graphs form a pullback diagram. We consider a larger
category of relative graphs that correspond to relative Toeplitz graph
algebras. In this setting we give necessary and sufficient conditions on the
pushout to get a pullback of $C^*$-algebras.
