Example [efr-00A1]

Consider a double category with four objects A,B,B',C, two nonidentity vertical maps B \to B' \to B (which necessarily form an isomorphism), and two nonidentity horizontal maps A \nrightarrow B, B' \nrightarrow C (and no nonidentity squares). If \mathbb {C}_0 is the vertical category here, there is an obvious equivalence to the discrete category on three objects \mathbb {C}_0 \to \{A,B,C\} (identifying B,B'). But if we take any equivalence \mathbb {C}_1 \xrightarrow {\sim } \mathbb {C}_1' extending this diagram:

we find that the two horizontal arrows have become composable, and so there cannot be any pseudo double category structure on the right-hand side.