Intuition for Coconstant morphisms A constant morphism $f \in \mathrm{Hom}(X,Y)$ is a morphism such that for any object $Z$ and any morphisms $g,h \in \mathrm{Hom}(Z,X)$, $f \circ g = f \circ h$. This is very easy to grasp and one can think of many examples. For instance, in the category of continuous real-valued functions, any map $f(x) = k$ where $k$ is a real number proves to be a constant morphism. 
On the other hand, a morphism  $f \in \mathrm{Hom}(X,Y)$ is coconstant if for any object $Z$ and morphisms $g,h \in \mathrm{Hom}(Y, Z)$ we have $g \circ f = h \circ f$. In this case, the only real-valued function that I can think of that satisfies this is the $0$-map. 
What are some good, nontrivial examples of coconstant functions that can provide intuition for understanding this concept? That is, other than the obvious use of this device to provide duality for the constant morphism, does it have any deeper meaning/significance? I understand what the definition means but don't really see its use beyond the simple examples given.
 A: Suppose that your category has an initial object $I$.
Then any composition $f:X\stackrel {u}{\to} I \stackrel {v}{\to} Y$ is co-constant .
Indeed if you have two morphisms $g,h:Y\to Z$ and if $i:I\to Z$ is the unique morphism, then:
$$g\circ f=g\circ v\circ u=i\circ u=h\circ v\circ u=h\circ f$$  In other words,  any morphism factorizable through $I$ is co-constant.
In particular any morphism with source $I$ is co-constant.
Example
In the category of commutative rings $ \mathbb Z$ is an initial object.
So, given a ring $A$,  the  morphism $f:\mathbb Z[X]\to A:P(X)\mapsto P(17)\cdot 1_A$ is co-constant since it is the composition $\mathbb Z[X]\stackrel {eval_{17}}\to \mathbb Z\to A:P(X)\mapsto P(17)\mapsto P(17)\cdot 1_A$
A: This is the intuition behind it:
In some contexts you can talk about constant functions (eg. set theory, linear algebra, topology, ect). Smart categorists generalized the idea to all categories with: constant morphism
In some other contexts you can talk about functions going to 0 or going to some neutral element (eg. linear algebra, group/ring theory, ect). Smart categorists generalized the idea to all categories with: coconstant morphism.
using category theory you discover that these ideas (constant function vs. function to 0) are dually related. Something that is not too obvious.
Also:
If a category has a terminal object, then any morphism going to the terminal object is a constant morphism and , guess what, categorists called the generic terminal object 1 (the archetipal constant).
If a category has an initial object, then any morphism coming  from the initial object is a coconstant morphism and , guess what, categorists called the generic initial object 0 (the archetipal neutral element).
So we have a happy circumstance of nice ideas supported by suggestive notation.
See also Zero morphism for more examples
A: Take a function $f:\{k\} \to Y$ for any real number $k$ and $Y \subseteq \mathbb{R}$
