Consider topological complex $K$-theory, in this case for a connected finite CW-complex $X$ we have the following $K(X)=[X,BU]$.
Naively I'd like to use the natural isomorphism $K(-)=[-,BU]$ to take the natural transformation, $$K(X) \to K(X)$$ $$a \mapsto -a$$ and obtain a map $BU \to BU$ however it is clear that $BU$ is not a finite CW-complex and in the literature it is stated that one should use a Milnor Exact sequence. Could someone develop or give me a reference for this argument?