  Using ambient pressure x-ray and high pressure neutron diffraction, we
studied the pressure effect on structural and magnetic properties of
multiferroic Mn$_{1-x}$Co$_x$WO$_4$ single crystals ($x=0, 0.05, 0.135$ and
$0.17$), and compared it with the effects of doping. Both Co doping and
pressure stretch the Mn-Mn chain along the $c$~direction. At high doping level
($x=0.135$ and $0.17$), pressure and Co doping drive the system in a similar
way and induce a spin-flop transition for the $x=0.135$ compound. In contrast,
magnetic ground states at lower doping level ($x=0$ and $0.05$) are robust
against pressure but experience a pronounced change upon Co substitution. As Co
introduces both chemical pressure and magnetic anisotropy into the frustrated
magnetic system, our results suggest the magnetic anisotropy is the main
driving force for the Co induced phase transitions at low doping level, and
chemical pressure plays a more significant role at higher Co concentrations.
