The paper presents recent developments in the previously proposed generalized approach to algebraic specification of distributed systems based on the novel category-theoretic construction called graphalgebra. The graphalgebraic specification is based upon a directed multigraph, the edges of which represent computational operations performed in the nodes of the system and the vertices denote the data exchange ports between the components. Thus, deployment of operations upon the system nodes is specified explicitly. It is also advisable to explicitly describe, in the language of graphalgebras, the procedures for constructing systems towards the target deployment. To this end, the paper defines the constructions of subgraphalgebra, quotient graphalgebra, and bisimulation of graphalgebras and proves their key properties for the first time. The means to construct limits and colimits of suitable diagrams of graphalgebras are proposed. The theoretical results are illustrated by an example of calculating a limit in the category of deep neural networks.