The aim of this work is the construction of a "supermanifold of morphisms X→Y", given two finite-dimensional supermanifolds X and Y. More precisely, we will define an object SC^∞(X,Y) in the category of supermanifolds proposed by Molotkov and Sachse. Initially, it is given by the set-valued functor characterised by the adjunction formula Hom(PxX, Y) ≅ Hom(P, SC^∞(X,Y)) where P ranges over all superpoints. We determine the structure of this functor in purely geometric terms: We show that it takes values in the set of certain differential operators and establish a bijective correspondence to the set of sections in certain vector bundles associated to X and Y. Equipping these spaces of sections with infinite-dimensional manifold structures using the convenient setting by Kriegl and Michor, we obtain at a supersmooth structure on SC^∞(X,Y), i.e. a supermanifold of all morphisms X→Y.