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.)