How should we think of maps to the intial object? A final object in a category is one that has a unique map from any other object. An intuitive way of thinking about the final object is as the 'point'. Then
we think of maps from the 'point' to another object as probing the point structure of that object.
How about the dual?
An initial object in a category is one that has a unique map to any other object. Is there a correspondingly intuitive picture of an initial object? Perhaps as a 'copoint' - whatever that could mean. This would mean of thinking about maps to the 'copoint'.
 A: Be careful with the notion of a point. If we define a point to be a morphism from the final object, then every group has just one point, which is kind of boring.
A morphism to an initial object is just a morphism from a final object in the dual category. Thus, formally, nothing new happens here. But this argument glosses over the fact that the categories of interest are not closed under duals (of course we can take the dual of any category, but I claim that often this is not an interesting one). So let us look at interesting examples.
Many "geometric" categories turn out to be extensive, and in an extensive category the initial object is strict, meaning that every morphism to it is an isomorphism. Examples include $\mathsf{Set}$, $\mathsf{Top}$, $\mathsf{Man}$, $\mathsf{Sch}$, $\mathsf{Met}$, $\mathsf{Pos}$.
The other extreme are categories with a zero object. Here every object has a unique morphism to the initial object, because it is final. This applies for example to $\mathsf{Ab}$ and to $\mathsf{Rng}$ (rngs).
Now look at $\mathsf{Ring}$, the category of rings (always with unit). The initial object is $\mathbb{Z}$. A homomorphism $R \to \mathbb{Z}$ is automatically surjective, hence is determined by its kernel. This is a subrng of $R$. Conversely, if $S$ is a rng, then we may consider its unitalization $\tilde{S} \in \mathsf{Ring}$ together with the canonical homomorphism $\tilde{S} \to \mathbb{Z}$. These constructions are inverse to each other and provide an equivalence of categories $\mathsf{Ring} / \mathbb{Z} \simeq \mathsf{Rng}$.
In general, if $C$ is a category with initial object $0$, then $C/0$ has a zero object $0:= (\mathrm{id} : 0 \to 0$) and is equipped with a forgetful functor $U : C/0 \to C$ mapping $0 \mapsto 0$. If $D$ is another category with a zero object $0$ and a functor $F : D \to C$ mapping $0 \mapsto 0$, then we define $\tilde{F} : D \to C/0$ via $d \mapsto (F(d) \to F(0)=0)$. Clearly $F=U\tilde{F}$, $\tilde{F}(0)=0$ and $\tilde{F}$ is unique with this property. This shows that $C \mapsto C/0$ is right adjoint to the forgetful functor from the (meta)category of categories with a zero object (and functors preserving them) to the (meta)category of categories with an initial object (and functors preserving them).
A: "Intuitive picture" is usually achieved through examples. So, here you have some.


*

*In the category of sets, the initial object is the empty set $\emptyset$ and the unique map $\emptyset \longrightarrow S$ to every set $S$ is the inclusion.

*In the category of groups, the initical object is the group with just one element $\{ e\}$ and the unique map $\{ e\} \longrightarrow G$ to every group $G$ is the one that sends $e$ to the neutral element of $G$.

*In the category of vector spaces, the initial object is the vector space with just one vector $\{\overrightarrow{0} \}$ and the unique map $\{\overrightarrow{0} \}\longrightarrow   V$ is the one that sends $\overrightarrow{0}$ to the zero vector in $V$.


Get the (co)point?  :-)
