  In the literature, one finds several competing notions for the super (i.e.,
Z/2-graded) analog of a monoidal category. The goal of this paper is to clarify
these definitions and the connections between them. We also discuss in detail
the example of the odd Temperley-Lieb supercategory. In a forthcoming article,
we will exploit the formalism developed here in order to define super analogs
of the Kac-Moody 2-categories of Khovanov-Lauda and Rouquier.
