The local algebras of the maximal Coset model C_max associated with a chiral conformal subtheory A\subset B are shown to coincide with the local relative commutants of A in B, provided A contains a stress energy tensor. Making the same assumption, the adjoint action of the unique inner-implementing representation U^A associated with A\subset B on the local observables in B is found to define net-endomorphisms of B. This property is exploited for constructing from B a conformally covariant holographic image in 1+1 dimensions which proves useful as a geometric picture for the joint inclusion A\vee C_max \subset B. Immediate applications to the analysis of current subalgebras are given and the relation to normal canonical tensor product subfactors is clarified. A natural converse of Borchers' theorem on half-sided translations is made accessible.