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 It seems that in some cases like domain/codomain such dual pairs superseded older notions (source/target) while others pairs survived in their classical form (mono/epi, initial/terminal or pull back/push out).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?