Connected categories and connected limits Let $C$ be a category together with an equivalence relation $∼$ on the objects by $x ∼ y$ whenever there is a morphism $f: x \to y$. 


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*What does $(Ob(C)/∼) \cong 1$ mean? What kind of isomorphism is this?

*What does it mean for a category to have all small connected limits?


First of all, note that this relation is in general not symmetric, so we have to take the equivalence relation generated by this relation. That is, $x\sim y$ if and only if there is a zigzag of morphisms $x\to x_1\leftarrow x_2\to \dots  \leftarrow x_n \to y$ in $C$. Therefore, we can think of an equivalence class as a connected component of $C$; it's a bunch of objects which can be joined by a zigzag of morphisms. Consequently, if $\mathrm{Ob}(C)/{\sim} \cong 1$, meaning that there is precisely one equivalence class, it makes sense to call $C$ connected: There exists an object of $C$ and each pair of objects can be connected through a zigzag of morphisms.
Now, a connected limit is one where the indexing category is connected and a category has all connected limits if all such connected limits exist.
