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We characterise the Whitney equisingularity of families of corank one mapgerms $f_t\colon(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolatedinstabilities in terms of the constancy of the $\mu_I^*$-sequences of $f_t$ andthe projections $\pi\colon D^2(f_t)\to\mathbb{C}^n$, where $D^2(f_t)$ is thedouble point space in $\mathbb{C}^n\times\mathbb{C}^n$. The $\mu_I^*$-sequenceof a map germ consist of the image Milnor number (in Mond's sense) of the mapgerm and all its successive transverse slices. Since $D^2(f)$ is an ICIS in$\mathbb{C}^n\times\mathbb{C}^n$ (when $f$ has corank one and has isolatedinstability), we need to develop the notions of image Milnor number and theimage computing spectral sequence for map germs defined on ICIS.

Whitney equisingular unfoldings of corank 1 germs