The authors extend the notion of $\phi$-strict minimizer to set-valued maps, in such a way that the concept of a $\phi$-strict minimizer, presented in {\it E. M. Bednarczuk} [Optimization 53, No. 5--6, 455--474 (2004;

Zbl 1153.90529)] for vector functions, as well as the notion of a minimizer of order one introduced for set-valued maps in {\it G. P. Crespi, I. Ginchev} and {\it M. Rocca} [Math. Methods Oper. Res. 63, No. 1, 87--106 (2006;

Zbl 1103.90089)] are generalized in a unified manner. A structure theorem is proved for a vector-valued function which states that a point is a $\phi$-strict minimizer for a vector valued function $f$ over a set if and only if the point is a $\phi$-strict minimizer for a family of scalar functions and sets, each of these functions being the composition of $f$ with a positive, continuous and linear functional, and the family of sets, a covering of the initial set. Different kinds of strict minimizers for the set-valued problem are characterized through different kinds of strict minimizers for an associated scalarized problem. Finally, several optimality conditions are established for strict minimizers of order one. A characterization is given for a global minimizer through the radial derivative. Comparisons with other results are made, and some illustrative examples are provide.