Temporal cavity solitons, relying on the double balance of nonlinearity and dispersion as well as pump and loss, are mainly investigated in the scalar Kerr resonators. Recent works concentrating on the polarization-dependent vectorial cavity report on the generation of scaler and vectorial cavity solitons with asymmetric and symmetric excitation. Moreover, cavity solitons corresponding to distinct spatial modes have been observed in the micro-resonator through the sideband modulation. Here we report on the generation of spatial multiplexing solitons in a monochromatically driven Kerr resonator. The schematic of the cross-transmission cavity is proposed to eliminate the impact of a large wave vector mismatch. In the cases of symmetric and asymmetric cross-transmission cavity, we can respectively obtain vectorial solitons and coexisted nonlinear states. Our work not only delivers more insights into spatial multiplexing nonlinear dynamics in Kerr passive cavity but also provides a feasible method for the generation of broadband spatial multiplexing laser sources.