= Solution
First work in the receiver's <rest frame>. If a source recedes at <speed> $\beta c$, two successive emissions separated by source <proper time> $T_0$ occur $\gamma T_0$ apart in this frame. During that interval the source retreats by $\beta c\gamma T_0$, so the arrival interval is $T_R=\gamma T_0(1+\beta)$. The longitudinal <relativistic Doppler effect> is consequently
$$
\boxed{\frac{\nu_R}{\nu_0}=\sqrt{\frac{1-\beta}{1+\beta}},\qquad
\frac{\lambda_R}{\lambda_0}=\sqrt{\frac{1+\beta}{1-\beta}}.}
$$
A negative $\beta$ describes approach and gives a blueshift.
Interpret $633\,\mathrm{nm}$ as the laser's rest-frame wavelength and the stated detector <velocity> as measured in the initial laboratory frame. Choose $+x$ from source to sensor, with detector <velocity> $-0.2c$. The required relative recession <speed> follows from
$$
q=(1000/633)^2,\qquad\beta_{\rm rec}=\frac{q-1}{q+1}=0.4278687\ldots.
$$
By <relativistic velocity addition>, the source <velocity> $\beta_Sc$ in the laboratory obeys $(\beta_S+0.2)/(1+0.2\beta_S)=-\beta_{\rm rec}$. Hence
$$
\boxed{\beta_S=-\frac{\beta_{\rm rec}+0.2}{1+0.2\beta_{\rm rec}}=-0.578375\ldots.}
$$
The laser must move \b[away from the sensor at about $0.5784c$ in that laboratory frame]. Although the sensor moves toward the source's initial location, the retreating source produces the required net redshift. The frame qualification matters: a <velocity> permanently fixed relative to the moving source would specify the Doppler factor already, rather than leave it adjustable.
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