  We use Kirk's invariant of link maps $S^2\sqcup S^2\to S^4$ and its
variations due to Koschorke and Kirk-Livingston to deduce results about
classical links. Namely, we give a new proof of the Nakanishi-Ohyama
classification of two-component links in $S^3$ up to $\Delta$-link homotopy. We
also prove its version for string links, which is due (in a slightly different
form) to Fleming-Yasuhara. The proofs do not use Clasper Theory.
