Lurie defines $\mathfrak{C}:Set_\Delta \rightarrow Cat_\Delta$ the functor from simplicial sets to simplicially enriched categories. And defines:
Simplicial sets $X$ are considered to be categorically equivalent to $Y$, if $\mathfrak{C}[X] $ and $\mathfrak{C}[Y]$ is an equivalence of $H$ enriched category.
Here $H$ denotes the homotopy category of simplicial sets. Does this notion commute with colimit? I.e. if $\{X_\alpha \rightarrow Y_\alpha\}$ are all categorically equivalent, then the induced map on their colimit is categorically equivalent.
This may seem like a direct consequence of the fact that $\mathfrak{C}$ is a left adjoint : but only of the underlying category. So this doesn't seem to apply.
My main concern is how Lurie shows that class of "covariant equivalences" is perfect. Or that of push out preserves covariant equivalences. (which is a special type of categorical equivalence). These are 2.1.4.6, and 2.1.4.7.
Categorical equivalences are the weak equivalences in a model structure. Such a class is essentially never closed under colimits. It is closed under homotopy colimits, and it is an important technical activity to describe classes of ordinary colimits which coincide with the corresponding homotopy colimits. A simple example is that a pushout of two cofibrations between cofibrant objects is always a homotopy pushout.