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Dynamic reconfiguration, understood as the ability to manage at run time the live components and how these interact in a system, is a feature that is crucial in various languages and computing paradigms, in particular in object orientation. In this paper, we study a categorical approach for characterising dynamic reconfiguration in a logical specification language. The approach is based on the notion of institution, which enables us to work in an abstract, logic independent, setting. Furthermore, our formalisation makes use of representation maps in order to relate the generic specification of components (e.g., as specified through classes) to the behaviour of actual instances in a dynamic environment. We present the essential characteristics for dealing with dynamic reconfiguration in a logical specification language, indicating their technical and practical motivations. As a motivational example, we use a temporal logic, component based formalism, but the analysis is general enough to be applied to other logics. Moreover, the use of representation maps in the formalisation allows for the combination of different logics for different purposes in the specification. We illustrate the ideas with a simple specification of a Producer-Consumer component based system.