From this comment on this MO question about locally Cartesian closed subcategories of topological spaces I understand the following proposition holds.
Proposition. Let $\mathsf C$ be the category of $k$-spaces. If $A,B$ are compactly generated, then $f^\ast :\mathsf C_{/B}\to \mathsf C_{/A}$ admits a right adjoint.
The comment directs to May-Sigurdsson, but it's terse for me and I'm struggling to find and isolate the statement and its proof.
(Remark 7.1.5.4 in Higher Topos Theory also mentions something about this.)
Is the proposition above true as it's written? How to prove it? When does base change along $\mathrm{eval}_0:B^I\to B$ admit a right adjoint?