  A rigorous limit procedure is presented which links nonlocal models involving
adhesion or nonlocal chemotaxis to their local counterparts featuring
haptotaxis and classical chemotaxis, respectively. It relies on a novel
reformulation of the involved nonlocalities in terms of integral operators
applied directly to the gradients of signal-dependent quantities. The proposed
approach handles both model types in a unified way and extends the previous
mathematical framework to settings that allow for general solution-dependent
coefficient functions. The previous forms of nonlocal operators are compared
with the new ones introduced in this paper and the advantages of the latter are
highlighted by concrete examples. Numerical simulations in 1D provide an
illustration of some of the theoretical findings.
