A simplicial category is a category enriched over the monoidal category of simplicial sets (morphism sets are now simplicial sets), and the collection of all such categories forms a category itself (modulo set theoretic issues). It is asserted on p. 23 of "Higher topos theory" that this latter category has all small colimits---why? How are such colimits computed?
I can certainly understand coproducts (take a disjoint union of everything in sight), but how does one construct coequalizers? In fact, I think I don't understand the latter even in the case of ordinary categories (to which one may perhaps reduce because a simplicial category is just a simplicial object in the category of categories).
For instance, consider the category $C_0$ that has a single object $*$ with no nonidentity morphisms, and consider the category $C_1$ which has four objects and two nonidentity morphisms $a \rightarrow b$ and $c \rightarrow d$. Consider the two functors $C_0 \rightarrow C_1$ that send $*$ to $b$ and $c$, respectively. What is the resulting colimit category in this case and why?