When reading through [Loregian and Riehl - Categorical notions of fibration](https://arxiv.org/abs/1806.06129), 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.



Why aren‘t op and co switched?
The problem is that for a long time there were only 1-categories and hence only one kind of duality, and sometimes people called it "op" and sometimes "co". Colimits and cofibrations were therefore the only possible notion of dual for limits and fibrations, and take place in the opposite category, which was also the only notion of dual for a category, and denoted "op" as being short for "opposite". Sometimes both "op" and "co" terminologies are in use for the same thing, e.g. "oplax" and "colax".


One should not attempt to remedy this by departing from the standard definitions of $C^{\rm op}$ and $C^{\rm co}$ for 2-categories. Among other reasons, they align with standard notions of duality for enriched categories: for a general $V$ there is only one notion of opposite $V$-category, denoted $C^{\rm op}$, and when a 2-category is regarded as a $\rm Cat$-enriched category this produces the 1-cell dual.


I think the most satisfying perspective on this (though not fully satisfying) is that the 2-cell dual *of* a 2-category, $C^{\rm co}$, corresponds to classical dual constructions (often denoted by "co") *on* its objects, regarded as generalized categories. For instance, while a comonad *on* a category $A$ is equivalently a monad on the category $A^{\rm op}$, a comonad *in* a 2-category $C$ (on an object $A\in C$) is equivalently a monad in $C^{\rm co}$. (A monad in $C^{\rm op}$ is just the same as a monad in $C$.)


