Possible semantics for categorical  co-constness 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.
 A: Edit: This is not quite right -- the two definitions of constantness are not equivalent even in the category of sets, which means that the representability argument doesn't quite work.  See the nlab page for details, but note that they are equivalent if the hom sets $C(X,A)$ are inhabited for every X, or equivalently if A admits a global section.
This doesn't answer all of your questions, but...
A morphism is constant in your sense iff it is representably constant in (what you say is) Lawvere's sense, that is if $k_* \colon C(X,A) \to C(X,B)$ factors through the terminal object in Set.  So if C itself has a terminal object then the two are equivalent.
A morphism c is co-constant iff $c^* \colon C(B,Y) \to C(A,Y)$ is constant, and it's easy to see that if C has an initial object then it's equivalent to ask that c factor through it.  But in Set the initial object $\emptyset$ is strict, meaning that any morphism into $\emptyset$ is an isomorphism.  So there are no non-trivial co-constant morphisms in Set, or indeed in any category with a strict initial object (such as (by a result of Joyal) any cartesian closed category with initial object, like a topos).
I can't tell off the top of my head whether your definition would be useful elsewhere, but the above does put some restrictions on what sorts of categories non-trivial co-constant morphisms might turn up in.
