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'.

"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?  :-)
