This is one of those "does such-and-such have a name" and "where can I read about it" emails. I'm interested in a particular map F : A -> B between strict 2-categories. The codomain is something like Cat and the domain has exactly two objects, say 0 and 1. If we restrict to the sub 2-category at 0, then F is a lax functor with 2-cells for both composition and the identity; if we restrict to 1, then F is oplax. In general, there is a 2-cell in the usual square whose boundary encodes functoriality with respect to composition of 1-cells and that 2-cell points in the lax direction if the composition occurs at 0 and the oplax direction if it occurs at 1. Up to this contradicting directionality, everything is "maximally coherent". For example, the usual pasting diagram for associativity of, say, a pseudofunctor is a diagram of 2-cells that sit inside two rectangles whose domain object is A(c,d) x A(b,c) x A(a,b) and whose codomain is B(Fa,Fd). If b=c then the 2-cells point in compatible directions and the pasted composites agree. If b and c differ, the 2-cells aren't composable directly but if we formally inverted one set then the diagram would commute. Any comments or suggestions would be appreciated. Thanks, Emily Riehl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]