Diagrammatics for lax and Frobenius monoidal functors and weak morphism classifiers

The theory of 2-monads entails that, for a strict monoidal category C, there is a strict monoidal category L© such that strict monoidal functors from L© are precisely the lax monoidal functors from C. We give an elementary, diagrammatic, construction of L© and of its variants for oplax and Frobenius lax functors. The diagrams used are analogous to the diagrammatics for lax monoidal functors studied by McCurdy.