We define the affinization of an arbitrary monoidal category C, corresponding to the category of C-diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to C. The affinization formalizes and unifies many constructions appearing in the literature. In particular, we describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants. When C is rigid, its affinization is isomorphic to its horizontal trace, although the two definitions look quite different. In general, the affinization and the horizontal trace are not isomorphic. We then use the affinization to show our main result, which is an explicit isomorphism between the central charge k reduction of the universal central extension of the elliptic Hall algebra and the trace, or zeroth Hochschild homology, of the quantum Heisenberg category of central charge k. We use this isomorphism to construct large families of representations of the universal extension of the elliptic Hall algebra.