  This paper studies the asymptotic behavior of coexistence steady states of
the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend
to infinity at the same rate. In the case when either one of two
cross-diffusion coefficients tends to infinity, Lou and Ni (1999) derived a
couple of limiting systems, which characterize the asymptotic behavior of
coexistence steady states. Recently, a formal observation by Kan-on (2020)
implied the existence of a limiting system including the nonstationary problem
as both cross-diffusion coefficients tend to infinity at the same rate. This
paper gives a rigorous proof of his observation as far as the stationary
problem. As a key ingredient of the proof, we establish a uniform $L^{\infty}$
estimate for all steady states. Thanks to this a priori estimate, we show that
the asymptotic profile of coexistence steady states can be characterized by a
solution of either of two limiting systems.
