I am a little bit confused by Serre and Hurewicz fibrations in the context of pointed spaces, i.e. in **$Top_*$**.

Serre and Hurewicz fibrations are defined in **$Top$**, i.e. for non-pointed spaces, as having the RLP for inclusions $i_0 \colon I^n \hookrightarrow I^n \times I$ and $i_0 \colon X \hookrightarrow X \times I$ respectively.

But if I consider now a pointed map $p \colon (E,e_0) \to (B,b_0)$ and a pointed homotopie $H \colon (I^n,\partial I^n) \times I \to (B,b_0)$ for Serre or $H \colon (X,x_0)\times I \to (B,b_0)$ for Hurewicz, and if $p$ is a Serre/Hurewicz fibration in Top, there is no garantee that the lift $\tilde{H}$ is also a pointed homotopy. All we can say is that $\tilde{H}(* \times I) \subset F$ where $F = p^{-1}(b_0)$ is the fiber of $p$. So a priori $p$ **is not** a Serre/Hurewicz fibration in $Top_*$

Thus it seems that a specific more restrictive notion of Serre/Hurewicz fibration should be used in the context of $Top_*$, as having the RLP in $Top_*$ for the mentionned inclusions, i.e. the lift is should also be a pointed homotopy.

But I have never seen explicitely such a definition in classic books or pdf, although in most theorems involving homotopy groups we are in fact in the $Top_*$ context. Why ?

Now it is true that Serre fibration also have the RLP for $i \colon J^n \hookrightarrow I^n \times I$ where $J^n = \partial I^n \times I \cup I^n \times 0$. If there is a commutative square as usual where the initialization is $J^n \mapsto e_0$, then the lift is such that $\tilde{H}(\partial I^n \times I) = e_0$ and $\tilde{H}$ is a pointed homotopy. In this way, a Serre fibration is also a "pointed" Serre fibration. But this does not work for Hurewicz.