Let $B$ be a topological space. Consider the evaluation at zero of paths in $B$. This is a continuous map $\operatorname{ev}_0:B^I\to B$ where the domain carries the compact-open topology.
For which spaces $B$ does the pullback functor $\mathsf{Top}_{/B}\overset{\operatorname{ev}_0^\ast}{\longrightarrow}\mathsf{Top}_{/B^I}$ have a right adjoint? Is local connectedness enough?
(If it's more convenient - same question with $\mathsf{Top}$ replaced by the full subcategory of locally connected spaces.)
Added. Perhaps my question is overly general. Here's what I am specifically looking for.
Let $s$ be a Hurewicz connection on a bundle $\begin{smallmatrix}A\\ \;\;\downarrow \alpha\\ B \end{smallmatrix}$. By definition $s$ is a bundle map $\operatorname{ev}_0^\ast \alpha \to \alpha^I$. I am wondering if there is some sort of bundle over $B$, worthy of being called $\Pi_{\operatorname{ev}_0}\alpha^I$, such that the Hurewicz connection uniquely corresponds to a bundle arrow $s_\flat:\alpha\to \Pi_{\operatorname{ev}_0}\alpha^I$. In other words, given a Hurewicz connection, can we (continuously!) separate its input into "path" and "initial point of lift" portions?
