Taking the homotopy pullback always results in fibrations? My question is quite simple, if we have a diagram $X \rightarrow Z \leftarrow Y$ in the category of topological sapces, then when taking the homotopy pullback we get a homotopy commutative diagram \begin{alignat}{9} P&\rightarrow\ &X \\ \downarrow & &\downarrow \\ Y &\rightarrow &Z \end{alignat}
My question is whether $P\rightarrow X$ and $P\rightarrow Y$ are always fibrations.
I think this is true because one equivalent way of obtaining a homotopy pullback is to first take for example $X \rightarrow Z$ and write it as a fibration $E \rightarrow Z$ ($E$ is of the same homotopy type as $X$), and then take the conventional pullback of $E \rightarrow Z \leftarrow Y$ \begin{alignat}{9} P&\rightarrow\ &E \\ \downarrow & &\downarrow \\ Y &\rightarrow &Z \end{alignat} then since $E \rightarrow Z$ is a fibration $P \rightarrow Y$ is also a fibration (common known propety of pullbacks). And one may substitute $Y\rightarrow Z$ by aa fibration, to conclude that $P\rightarrow X$ is a fibration.
So my question is if this implies that any choice of $P$, $P \rightarrow X$ and $P \rightarrow Y$ will be so that $P \rightarrow X$ and $P \rightarrow Y$ are fibrations. If this is not true, is it atleast true for the standard construction? Where one takes $$P=\{(x,\gamma,y)\in X\times Z^I\times Y\ |\ \gamma(0)=x\ \text{and}\ \gamma(1)=y)\} $$ and $P \rightarrow X$ and $P \rightarrow Y$ are simply the projections.
 A: The double mapping path space $P=P_{f,g}$ associated with a cospan
$$X\xrightarrow{f}Z\xleftarrow{g}Y$$
is obtained as the strict pullback of the cospan
$$X\times Y\xrightarrow{f\times g}Z\times Z\xleftarrow{e_{0,1}} Z^I$$
where $Z^I$ is the space of maps $I\rightarrow Z$ in the compact-open topology. Here $e_{0,1}$ is the fibration given by the endpoint evaluation $e_{0,1}(\ell)=(\ell(0),\ell(1))$.
Thus
$$P_{f,g}\cong \{(x,\ell,y)\in X\times Z^I\times Y\mid f(x)=\ell(0),\;g(y)=\ell(1)\}.$$
Since fibration are stable under pullback we obtain a fibration,
$$\pi:P_{f,g}\rightarrow X\times Y,\qquad (x,\ell,y)\mapsto (x,y)$$
as the pullback of $e_{0,1}$. Since a composition of two fibrations is another fibration, we obtain fibrations
$$\pi_X:P_{f,g}\rightarrow X,\qquad \pi_Y:P_{f,g}\rightarrow Y$$
as the composites of $\pi$ with the projections
$$X\leftarrow X\times Y\rightarrow Y.$$
Note that there is nothing to be said about the general case: the square
\begin{array}{ccc}
\ast &\rightarrow & \mathbb{R} \\
\downarrow & &\ \downarrow \\
\mathbb{R} &\rightarrow& \mathbb{R}
\end{array}
is a homotopy pullback.
