We introduce a subcategory of directed graphs for which the construction of graph C*-algebras induces a contravariant functor into the category of algebras. Then we prove a theorem stating under which conditions this functor turns pushouts of directed graphs into pullbacks of algebras. To broaden the scope of pullback theorems, we also consider a covariant induction of morphisms between graph C*-algebras given by specifically tailored path morphisms of graphs that substantially enlarge the standard category of graphs by dropping the requirement that morphisms preserve the length of paths. For instance, the natural *-homomorphism from the minimal unitization of compact operators to the Toeplitz algebra (shrinking the boundary of the Klimek-Lesniewski quantum disc to obtain the standard Podleś quantum sphere) is covariantly induced by a morphism mapping edges to ever longer paths. The goal of this talk is to unravel how to define morphisms of graphs to maximize the applicability of functorial inductions from categories of graphs into the category of graph C*-algebras. Based on joint work with Mariusz Tobolski.