(co)limits do not respect homotopy equivalence
For example, we can regard $S^n$ as the colimit $$D^n\sqcup_{S^{n-1}}D^n\cong S^n.$$ Now $D^n\simeq \ast$, but $$\ast\sqcup \ast\not\simeq S^n.$$
For example, we can regard $S^n$ as the colimit $$D^n\sqcup_{S^{n-1}}D^n\cong S^n.$$ Now $D^n\simeq \ast$, but $$\ast\sqcup \ast\not\simeq S^n.$$