JSF:Facelet:Composite Component:Troubleshooting: Cannot find component with id blah

Wrong 1:

<composite:implementation>
 <p:inputText id="input" />
 <p:message for="#{cc.clientId}:input" />

Wrong 2:

<composite:implementation>
 <p:inputText id="input" />
 <p:message for="#{cc.component}:input" />

Right:

<composite:implementation>
 <p:inputText id="input" />
 <p:message for="input" />