In this talk I will describe certain categories of graphs and suitable morphisms that admit either a covariant or a contravariant functor into graph C*-algebras. Our main categories of interest are graphs with admissible graph homomorphisms AGH and graphs with admissible path homomorphisms APH. The former category admits a contravariant functor into C*-algebras and the latter category admits a covariant one. I will show examples of morphisms in each of these categories. In the case of AGH, I will talk about (generalized) out-splits defined by Bates and Pask. For the category APH, I will talk about morphisms induced by (generalized) shift moves. The main advantage of considering both categories is that many isomorphisms of C*-algebras induced by moves can be realized using these functors. Time permitting, I will end the talk by illustrating how isomorphisms given by multiple different moves can be decomposed into morphisms induced by both functors. This talk is based on a joint work with G. G. de Castro, P. M. Hajac and E. A. Pacheco.