We study the problem of deriving a specification for a third-party component, based on the specification of the system and the environment in which the component is supposed to reside. Particularly, we are interested in using component specifications for conformance testing of black-box components, using the theory of input-output conformance (ioco) testing. We propose and prove sufficient criteria for decompositionality, i.e., that components conforming to the derived specification will always compose to produce a correct system with respect to the system specification. We also study the criteria for strong decomposability, by which we can ensure that only those components conforming to the derived specification can lead to a correct system.