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.
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.