We define the affinization of an arbitrary monoidal category $\mathcal{ C}$, corresponding to the category of $\mathcal{ C}$-diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to $\mathcal{ C}$. 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 $\mathcal{ C}$ 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.
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