The Z/2-equivariant Heegaard Floer cohomlogy of the double cover of S^3 along a based link, defined by Lipshitz, Hendricks, and Sarkar, is an isomorphism class of F_2[\theta]-modules. In this talk, we show that, in the case of knots, this invariant is natural, and is functorial under based cobordisms. Then, as a topological application, given a transverse knot K in the standard contact 3-sphere, we will define an element of the Z/2-equivariant Heegaard Floer cohomology that depends only on the transverse isotopy class of K, and show that it satisfies some interesting properties.