Two diagrams $D,D'$ in a category $C$ (e.g.: directed systems, sub-posets, … ) are cofinal if they have equivalent colimits.
In most cases where the word “cofinal” is used, it seems to be the case $D'$ is a subdiagram of $D$ in whatever sense of subdiagram appears suitable. In that case the cofinality is equivalent to the inclusion functor being a cofinal functor.
As defined above, no relation is posited between $D$ and $D'$, and so it seems not too evil to define cofinalness^{1} as “a single object is a colimit for both diagrams”.
It is not said that they are final if they have equivalent limits — the “co”s are not freely mutable, although the dual situation is doubtless just as interesting.
Let $D' = D F$, and let $L\colon D \times (0 \to 1) \to C$ be an initial cocone over $D$; it seems natural to ask that $L \circ (F \times (0 \to 1))$ be an initial cocone over $D'$.
Nonempty subsets of a finite total order are cofinal iff they have the same maximum.
Every infinite subset of $\omega$ is cofinal with $\omega$, as diagrams in $\omega + 1$.
It is often said that two diagrams are cofinal even when neither has a colimit, if they acquire a common colimit on passing to a suitable completion of $C$. This can probably be phrased internally to $C$, at the cost of intuition.
cofinal functor, cofinal diagram
Cofinality is an ordinal invariant of ordinals, which doesn’t make sense in the present generality, so we need another name for the adjective. ↩