Complex K-theory: Inducing natural transformations to $BU$ 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?
 A: This is probably too late to help, but for posterity, I will answer this.
Let $\tilde{K}(-)$ denote the reduced K-Theory functor, which is representable by $BU$, the colimit of the infinite Grassmannians. Then by naively applying Yoneda, one gets the bijection $$\mathcal{N}at(\tilde{K}^n(-),\tilde{K}(-))=[BU^n,BU],$$
where by $\tilde{K}^n(-)$ I mean the $n$ fold product of the $\tilde{K}(-)$ functor (I include it incase one wants to induce $n$-ary operations on $BU$).
However, as you noticed this bijection might fail, since there are natural transformations in K theory which rely on the spaces being compact (equivalently a finite CW complex)- an example being the Adams operations $\{\psi^k\}_{k\in \mathbb{N}}$, which rely on the splitting principle for their construction. However as it turns out, the above naive equation will still work in the case of $BU$.
Suppose $X$ is a (possibly non-compact) space. Then $X=\varinjlim_i X_i$, where the colimit is over finite subcomplexes of $X$. Now recall that one has a Milnor exact sequence for $K$ theory, (and essentially any other generalised cohomology theory),-
$$0\to \varprojlim_{i} ^1[\Sigma X_i, BU]\to [X,BU]\to \varprojlim_i [X_i,BU]\to 0 ,$$
In general for a non-compact $X$ the $\varprojlim ^1 $ term will not vanish. However as it turns out, from a general result of Atiyah (see his book on K-Theory), a cell complex with cells only in even degrees has vanishing $K^1$. In particular $BU=\varinjlim_n BU(n)$, and where each $BU(n)$ is a colimit over $k$ of Grassmanians of $n$ planes in $n+k$ space. Now the Grassmanians can be given a cell structure with cells only in even degrees (see Milnor Stasheff) so one can write $BU=\varinjlim_i X_i$ for some finite $X_i$ and where each $X_i$ has cells in even degree. Then one has $$\varprojlim_{i}^1\tilde{K}(\Sigma X_i)= \varprojlim_{i}^1[\Sigma X_i,BU]=\varprojlim_{i}^1 K^{1}(X_i)=0,$$ where I just used the definiton of $K^1$. 
Thus one is left with an isomorphism $$[BU,BU]\cong \varprojlim_i[X_i,BU].$$
Therefore given an operation on $\tilde{K}(-)$ which depends on the compactness of $X$, one may still get a natural operation on $\tilde{K}(BU)$ by the above argument, and thus use the naive Yoneda lemma above. The same argument works for $n$-ary operations. In particular it works for the operation you mention in your question.
