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In this paper, we give some examples which point to the non-existence of $C^{\infty }$-global stable diagrams $R\stackrel{g}{\leftarrow }M\stackrel{f}{\to }R$, $M$ compact. If $\Phi$ : $M\to Q$ is fixed we define the $\Phi$-equivalence for maps $f:M\to P$ and the corresponding $\Phi$-stability. The globalization procedure works and we can compare the $\Phi$-stability, $\Phi$-infinitesimal stability, and $\Phi$-homotopical stability. Also we give some characterization theorems for lower dimensions.