A structure closely related to the one which Francois Lamarche asked about appeared in my thesis (1992) [see below] as a simple example of a sesqui-category which is not a 2-category. I too would expect it's appeared elsewhere, but I don't know where. John Stell \subsubsection{An Example of a Sesqui-Category} We include an example to show that there are naturally occurring sesqui-categories other than in connection with modelling term rewriting. The underlying category is {\bf Graph}. Suppose there are graphs $G$ and $H$, and graph morphisms $g,h : G \rightarrow H$. In this situation, the 2-cells $\alpha : g \rightarrow h$ are assignments to each node $n$ of $G$ of a path of edges from $ng$ to $nh$ in $H$. The compositions $\circ_R$ and $\circ_L$ are readily defined. If $f : F \rightarrow G$ then $f \circ_R \alpha$ assigns to a node $m$ of $F$ the path $(mf)\alpha$ in $H$. For the left composition, suppose we have $k : H \rightarrow K$. Since $n\alpha$ is a path in $H$, we obtain a path $(n\alpha)k$ by applying $k$ to each of the edges in the path $n \alpha$. Thus we define $\alpha \circ_L k$ to be the assignment to $n$ of the path $(n \alpha)k$. The vertical composition is the usual concatenation of paths. The identity 2-cells are assignments of zero length paths.