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)



pullback and fiber sequence
Yes. Here are some details.


1. 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.
2. 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 $D$.)
3. 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.


