In a cartesian bicategory$\mathbf{B}$, the pair of terms diagonal, codiagonal refer to the canonical comultiplication $\Delta: X \to X \otimes X$ and the dual multiplication $\nabla = \Delta_*: X \otimes X \to X$ on any object. While the comultiplication is not a true diagonal (because $\otimes$ is not a cartesian product in $\mathbf{B}$), it is the diagonal when seen as belonging to the subcategory of maps (left adjoints), where the restriction of $\otimes$ to $Map(\mathbf{B})$ becomes a 2-product. Similarly, $\nabla$ is not a true codiagonal on $\mathbf{B}$, but it becomes a codiagonal in the sense above when seen as belonging to $Map(\mathbf{B})^{op}$, the opposite obtained by reversing $1$-cells but not $2$-cells.

Possible confusion

The term ‘codiagonal’ is also sometimes used in the context of the theory of bisimplicial sets. For this use see total simplicial set, within that entry.