pullback and fiber sequence Let $A\rightarrow D\leftarrow C$ a diagram of connected pointed toplogical space where $A\rightarrow D$ is a fibration. Denote $P=A\times_{D}C$.  We obtain a homotopy fiber sequence $$ \Omega D\rightarrow P\rightarrow A\times C $$
If we suppose that $D=\Omega X$ for some pointed topological space $X$. Do we obtain a homotopy fiber sequence $$ P\rightarrow A\times C\rightarrow D ?$$
where the map $A\times C\rightarrow D$ is obtained as a composition $A\times C\rightarrow D\times D\rightarrow D$ (the second map is a concatenation of loops)
 A: Yes.  Here are some details.


*

*The space $P$ sits in  homotopy pullback diagram
$\require{AMScd}$
$$
\begin{CD}
P @>>> D \\
@VVV@VVV \\
A\times C @>>> D\times D
\end{CD}
$$
where the the right vertical map is the diagonal. In fact, one can see this by replacing the latter map with the free path fibration $D^I \to D\times D$. After this replacement the diagram becomes a pullback and a pullback with one of the terminal maps a fibration is always a homotopy pullback.

*When $D = \Omega X$, there the diagonal is induced by the map $m:D\times D \to D$ given by $(\gamma_1,\gamma_2) \mapsto \gamma_1 \cdot \bar\gamma_2$, where the bar means loop inversion.  This means that there is a commutative homotopy pullback
$$
\begin{CD}
D @>\text{diag} >> D\times D \\
@VVV @VV m V \\
* @>>> D
\end{CD}
$$
where $*$ is some contractible space. (Perhaps the easiest way to see this is to note that the diagonal $X\to X \times X$ has homotopy fiber $D$.)

*The base change of a map which is induced in also induced: this means that there is a commutative homotopy pullback
$$
\begin{CD}
P @>>> A \times C \\
@VVV @VVV \\
* @>>> D
\end{CD}
$$
where the bottom map is the same as in the previous diagram.
