I am looking at the statment which says that replacing equivalent diagrams preserves colimits. This is 4.3.1.5(3), p262.
This seems intuitively true but I am unable to write it out rigorously. It would be nice if someone explains some details.
An equivalent question would be
Let $p,q:K \rightarrow C$ be two simplicial maps between simplicial sets. Suppose we have a natural transformation $\alpha:p \rightarrow q$. How doesd one naturally construct map $C_{p/} \rightarrow C_{q/}$?