Many notaions come in pairs, to mention the most classical example, sine/cosine are related by the the fact that $\sin(\alpha)=\cos(\beta)$ if $\alpha+\beta=\pi/2$, i.e., the angles are complementary. Already in this example, I don't know if there is any reason why $\sin$ is somehow the orginal notion and $\cos$ is the derived one. The wikipedia entry on trigonometric functions says The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle).

In category theory almost every notion has a dual which is usually called the co-notion (this is so common that it motivates many jokes like a co-mathematician is a mashine turning theorems into coffee or even rolarries into ffee -- found similarly on MO). It seems that in some cases like domain/codomain such dual pairs superseded older notions (source/target) while others pairs survived in their classical form (mono/epi, initial/terminal or pull back/push out).

The wikipedia entry for limits in category theory has a note on terminology where it is written the prefix "co" implies "first variable of the Hom''. I understand this for limits in the sense that they are universal with respect to morphisms into the limit while colimits are universal for morphisms out of. But I do not understand how this applies to codomain.

There are other notions which both come with a prefix like covariant/contravariant (here, as it seems, the prefixes are meant as with/against), left adjoint/right adjoint (okay, not exactly prefixes) or projective/inductive limit (which is criticized by the mentiond wikipedia page as a source of a lot of confusion).

Is there a general rule (at least of thumb) for a dual pair of notions A/B in category theory (i.e, B is A in the opposite category), which of them is called P so that the other is co-P?

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    $\begingroup$ In Mac Lane, the author notes on several occasions that the co- notation is used inconsistently in the literature for certain concepts. As far as I remember, this e.g. includes the question what is the adjoint and what is the coadjoint. This indicates that, even if there was a consistent naming convention, that this convention is not widely in use and that you thus have to learn what is co- and what not on a case to case basis (with maybe "local" consistency, e.g. everything directly derived from limits/colimits). $\endgroup$
    – NeitherNor
    Jul 26, 2020 at 14:18
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    $\begingroup$ For the trig functions, the "co-" ones are the ones decreasing for acute angles $\endgroup$ Jul 26, 2020 at 14:22
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    $\begingroup$ It always amuses me that "inverse limits" are "limits" and "direct limits" are "colimits". $\endgroup$ Jul 26, 2020 at 14:23
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    $\begingroup$ The general rule is that colimit-like things get the co- prefix... but this doesn’t always hold. For example, tensors (in the sense of enriched category theory) are colimit-like, whereas cotensors are limit-like. $\endgroup$
    – Zhen Lin
    Jul 26, 2020 at 15:41
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    $\begingroup$ @ZhenLin Which is probably the reason for revisionist efforts to use "copowers" and "powers" instead, with as far as I can tell only moderate success. $\endgroup$ Jul 26, 2020 at 16:53


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