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?



Colimits in the category of simplicial categories
To expand on Dmitri Pavlov's answer, the recipe for colimits, as in any monadic category, will be the following.


1. Take the colimit of the underlying (simplicial) graphs.
2. Apply the free functor.
3. Mod out by all relations that existed in the (simplicial) categories you're taking the colimit of.


Colimits in $\mathsf{Cat}$ are already notoriously bad (depending on the indexing category). For example, the coequalizer of the two inclusions $[0]^\to\_\to [1]$ is $\mathbf{B}\mathbb{N}$, the category with one object and an $\mathbb{N}$'s worth of endomorphisms. However, as you point out, coproducts are not so bad. Filtered colimits are likewise computed as on the underlying graphs. There is a [model structure on $\mathsf{Cat}$](https://ncatlab.org/nlab/show/canonical+model+structure+on+Cat) which gives you a notion of "homotopy colimit", and only those colimits which are actually homotopy colimits should be considered "good". Similarly, the Bergner model structure on simplicial categories can tell you which colimits of simplicial categories are good.


