We discuss a categorical construction of, and comparison of, the direct and inverse limits of a system of Banach spaces which is simultanously, and compatibly, directed ”upwards” by isometric inclusions, and ”downwards” by projections. This exercise in pure functional analysis provides a foundation for the infinitesimal deformation theory of inverse limit Riemann solenoids. Examination of the duality encountered reveals that the notion of ”cotangent space to Teichmuller Space” is somewhat subtle.