  We introduce two monoidal supercategories: the odd dotted Temperley-Lieb
category $\mathcal{T\!L}_{o,\bullet}(\delta)$, which is a generalization of the
odd Temperley-Lieb category studied by Brundan and Ellis, and the odd annular
Bar-Natan category $\mathcal{BN}_{\!o}(\mathbb{A})$, which generalizes the odd
Bar-Natan category studied by Putyra. We then show there is an equivalence of
categories between them if $\delta=0$. We use this equivalence to better
understand the action of the Lie superalgebra $\mathfrak{gl}(1|1)$ on the odd
Khovanov homology of a knot in a thickened annulus found by Grigsby and the
second author.
