Suppose that we have tree spaces $A, B, C$ (let say CW-complexes). Are given two pairs of maps:


$$f\_{0},g\_{0}: A\rightarrow B $$
$$f\_{1}, g\_{1}: A\rightarrow C $$
such that $f\_{0},g\_{0}$ are homotopic and $f\_{1},g\_{1}$ are also homotopic. 


Is it true that $colim [B\leftarrow^{f\_{0}} A\rightarrow^{f\_{1}} C] $
and $colim[B\leftarrow^{g\_{0}} A\rightarrow^{g\_{1}} C]$ are weakly equivalent ? 


Moreover, we assume that $g\_{0}= i\circ f\_{0}$ and $g\_{1}=j\circ f\_{1}$ where $i:B\rightarrow B $ and $j: C\rightarrow C$ are weak equivalences. 



pushout and homotopy
No. Take $A=[0,1]$, $B=S^1$, $C=\text{point}$. There is only one possible choice for $f\_1$ and $g\_1$. There are many possible choices for $f\_0$ and $g\_0$, but they are all homotopic. Choose $f\_0$ to be constant, and choose $g\_0$ to be surjective. Then the pushout for $f$ is $S^1$, and the pushout for $g$ is a point.


UPDATE:
Here is an adjusted answer for the adjusted question. Use the same spaces as before, but with the model $[0,1]/(0\sim 1)$ for $B=S^1$. Define $f\_0\colon A\to B$ by $f\_0(t)=[t/2]$, and define $g\_0$ by $g\_0(t)=[t]$. Define $h\_s\colon B\to B$ by $h\_s([t])=[\min(1,(1+s)t)]$, so $h\_0=1$ and $h\_1\circ f\_0=g\_0$. Again the pushout for $f$ is $S^1$, and the pushout for $g$ is a point.


