Let $X$ be a topological space. Is there a path object functor $\mathrm{P}:\mathbf{Top}\rightarrow \mathbf{Top}$, with fibration $ev_{0},ev_{1}: \mathrm{P}(X)\rightarrow X\times X$ such that:
(1) $\mathrm{P}$ is a cartesian functor i.e., $\mathrm{P}(X\times Y)$ is isomorphic to $\mathrm{P}(X)\times \mathrm{P}(Y).$
(2) and the concatenation map $-\ast-:\mathrm{P}(X)\times_{X}\mathrm{P}(X)\rightarrow \mathrm{P}(X)$ is strictly associative.
The condition (1) (but not (2)) is verified by the standard path object $\mathrm{P}(-)= \mathrm{Map}([0,1],-)$ and the condition (2) (but not (1)) is verified by the moore path object $\mathrm{M}(-)$ which is not a cartesian functor.