When reading through Loregian and Riehl - Categorical notions of fibration, on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^\text{op}$ to be the category $\mathcal C$ but with inverted $1$-cells and $\mathcal C^\text{co}$ to be the category $\mathcal C$ but with inverted $2$-cells.

Many constructions like for example limits and colimits seem to be named according to the co-notation but are mathematically op-dual. Similarly in the example above.

Hence the question is, whether there are any interesting constructions that are named “correctly” and also whether I am even correct about this feeling about things being switched up here.