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?

presentable(Def. A.1.1.2 in HTT) is the existence of small colimits, so invoking this more general property does not justify the existence of colimits. Concretely, without invoking more abstract nonsense, how does one describe the colimit in the example I give in the last paragraph? $\endgroup$nota simplicial object in categories (this is one of the reasons I dislike the name). $\endgroup$6more comments