I was just wondering whether there is already a name for an object of a category as described below.
Recall that an initial object in a category $\mathcal C$ is an object $I\in\mathcal C$ such that for all objects $C\in \mathcal C$ there exists exactly one morphism $I\to C$.
Now if I can express the category as the disjoint union of full subcategories $\mathcal C=\coprod_i \mathcal C_i$ and if $I_i$ is an initial object of $\mathcal C_i$ for some $i$, how would I call $I_i$?
In my situation it is equivalent to say that $I_i$ is an object of $\mathcal C$ such that no (non-identity) morphism has $I_i$ as a target. So I could call it a source, or a pure source (actually this more or less avoids the question rather than to solve it). Another option would be to say that $I_i$ is the initial object of a connected component of $\mathcal C$ motivated by the understanding of the graph or the classifying space of $\mathcal C$.
But I assume somebody must have considered something like that before.