If
B
G
,
B
H
{\mathbf {BG}},\;{\mathbf {BH}}
are the classifying spaces of compact Lie groups, with
H
{\mathbf {H}}
connected, then the mapping space functor
m
a
p
(
B
G
,
−
)
{\mathbf {map}}({\mathbf {BG}}, - )
commutes with
p
p
-completion on
B
H
{\mathbf {BH}}
: i.e., for each
f
:
B
G
→
B
H
f:{\mathbf {BG}} \to {\mathbf {BH}}
the component
(
m
a
p
(
B
G
,
B
H
)
f
)
p
∧
({\mathbf {map}}{({\mathbf {BG}},{\mathbf {BH}})_f})_p^ \wedge
is
p
p
-complete, and is homotopy equivalent to
m
a
p
(
B
G
,
B
H
p
∧
)
i
∘
f
{\mathbf {map}}{({\mathbf {BG}},{\mathbf {BH}}_p^ \wedge )_{i \circ f}}
.