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Largest Hausdorff quotient

Is there a left adjoint to ${\mathbf{Haus}}\to{\mathbf{Top}}$? Here ${\mathbf{Haus}}$ is the full subcategory of Hausdorff spaces in ${\mathbf{Top}}$?

If this is too much to ask, how about a left adjoint to the restriction to compact spaces ${\mathbf{CptHaus}}\to{\mathbf{CptTop}}$?