  In this paper we present $2$-category theory from the perspective of
Gray-categories using the graphical calculus of separated surface diagrams. As
an extended example we consider cones and limits of $2$-functors. Then we use
the canonical adjunction between $2$-computads and $2$-categories to interpret
the comparison structure of lax functors and extend the surface diagram
calculus with compositor sheets in order to represent and reason about them.
