Is there a way in which Conduche fibrations can lead to completeness in double categories? I know that Conduche conditions on functors play a role in completeness or cocompleteness in pseudo-double categories. (see http://www.numdam.org/article/CTGDC_2007__48_3_163_0.pdf)
Are there papers in which this idea is explored? (Struggling to find them!) Ayala has looked at this for $\infty-$categories.