In category theory a morphism is constant IIF it is absorbing (for left composition).
That is a morphism $k$ from $k:A\rightarrow B$ is constant if an only if for any two parrallel (same domain and same codomain) morphisms $f$ and $g$ of codomain $A$ we have $k(f) = k(g)$.
Now $c$ is co-constant IIF it is constant in the opposite category (equationally $f(c) = g(c)$ for any $f$ ...) .
Semantically (in Set at least) constness is rather clear , that is a non mathematician understands the idea of "unchanging", the question(s) is:
Q1: For constness are there other semantic appearing in something else than $Set$.
Q2 :What does it mean (semantically) to be coconstant in $Set$ and elsewhere ?
Note: I guess there are several (categorically prototypical) answers in both questions.
Remark : I thought of something like an unchanging measure of noise (constant) versus an intrinsically flat noise (coconstant). But this is a rather foggy intuition, moreover I cannot categorify it properly.