Diagramatics of A₁ × A₁ Folded Soergel Bimodules

There is an action of Z/2 on the category of Soergel Bimodules of type A₁ × A₁ induced by the nontrivial automorphism of its Dynkin diagram. We give an isotopy presentation by local generators and relations of the equivariantization of the category of Soergel Bimodules of type A₁ × A₁ under this action. This is the first step in a program to describe and study the categories of equivariant Soergel Bimodules.