In this paper we give a version of a conjecture of Lannes concerning the mod. $p$ cohomology of the space of maps from $B\mathbb {Z}/p\mathbb {Z}$ to a rather general space $X$. This gives a topological meaning to algebraic functor for modules over the Steenrod algebra, defined by Lannes. That functor has proven very useful in understanding spaces of maps from ifying spaces. As a corollary we get new proofs of several results of Lannes.