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Weakly globular double categories are a model of weak $2$-categories based onthe notion of weak globularity, and they are known to be suitably equivalent toTamsamani $2$-categories. Fair $2$-categories, introduced by J. Kock, modelweak $2$-categories with strictly associative compositions and weak unit laws.In this paper we establish a direct comparison between weakly globular doublecategories and fair $2$-categories and prove they are equivalent afterlocalisation with respect to the $2$-equivalences. This comparison sheds newlight on weakly globular double categories as encoding a strictly associative,though not strictly unital, composition, as well as the category of weak unitsvia the weak globularity condition.

Weakly globular double categories and weak units