A Serre fibration is a continuous map which possesses the homotopy lifting map against all disks. I think a Serre fibration moreover possesses the homotopy lifting property against all CW complexes, but I want to make sure whether my proof is correct and if it's not overkill.
Let $X$ be a CW complex. We want to lift a homotopy out of $X\times I$. We work by induction on the dimension, presenting the $n$-skeleton as a pushout of the $(n-1)$-skeleton with a coproduct of disks over their boundary. Now, because $I$ is exponentiable the functor $-\times I$ is a left adjoint and therefore preserves pushouts. Thus we can construct a lift using the universal property of pushouts, which then reduces us to the induction assumption and the HLP against disks.
Is this the correct argument, or are things purely formal and require no mention of exponentiability of the unit interval?
(After the comments) One can show that the map $D^n \to D^n\times I$ is homeomorphic to the map $D^n \cup S^{n-1}\times I \to D^n \times I$. Therefore, the latter maps can be used to show that $X\to X\times I$ is a relative $J$-cell complex if $X$ is a CW-complex. This argument does involve the exponentiability of $I$, unless you work in a convenient category where all objects are exponentiable. In short, I don't think the proof is only formal, and I think what you said is basically correct.