State-of-the-art hydrodynamical simulations show that gas inflow through the virial sphere of dark matter haloes is focused (i.e. has a preferred inflow direction), consistent (i.e. its orientation is steady in time) and amplified (i.e. the amplitude of its advected specific angular momentum increases with time). We explain this to be a consequence of the dynamics of the cosmic web within the neighbourhood of the halo, which produces steady, angular momentum rich, filamentary inflow of cold gas. On large scales, the dynamics within neighbouring patches drives matter out of the surrounding voids, into walls and filaments before it finally gets accreted on to virialized dark matter haloes. As these walls/filaments constitute the boundaries of asymmetric voids, they acquire a net transverse motion, which explains the angular momentum rich nature of the later infall which comes from further away. We conjecture that this large-scale driven consistency explains why cold flows are so efficient at building up high-redshift thin discs inside out.