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The motivic Hirzebruch class ${T_y}_*: K_0(\\m V /X) \\to H_*^{BM}(X) \\otimes \\bQ[y]$ is a unique natural transformation satisfying that for a nonsingular variety $X$ the value ${T_y}_*([X \\xrightarrow {\\op {id}_X} X])$ of the isomorphism class of the identity $X \\xrightarrow {id_X} X$ is the Poincar\\'e dual of the Hirzebruch cohomology class of the tangent bundle $TX$. 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