  We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$
and the classifying space by $BG$. Let $G$ be a well-pointed topological monoid
of the homotopy type of a CW complex and $G'$ a well-pointed grouplike
topological monoid. We prove the weak equivalence between the pointed mapping
space $\mathrm{Map}_0(B_nG,BG)$ and the space of all $A_n$-maps from $G$ to
$G'$. This fact has several applications. As the first application, we show
that the connecting map $G\rightarrow\mathrm{Map}_0(B_nG,BG)$ of the evaluation
fiber sequence
$\mathrm{Map}_0(B_nG,BG)\rightarrow\mathrm{Map}(B_nG,BG)\rightarrow BG$ is
delooped. As other applications, we consider higher homotopy commutativity,
$A_n$-types of gauge groups, $T_k^f$-spaces by Iwase--Mimura--Oda--Yoon and
homotopy pullback of $A_n$-maps. In particular, we show that the $T_k^f$-space
and the $C_k^f$-space are exactly the same concept and give some new examples
of $T_k^f$-spaces.
