Silicon nitride is an excellent material platform for its extremely low loss in a large wavelength range, which makes it ideal for the linear processing of optical signals on a chip. Moreover, the Kerr nonlinearity and the lack of two-photon absorption in the near infrared enable efficient nonlinear optics, e.g., frequency comb generation. However, linear and nonlinear operations require distinct engineering of the waveguide core geometry, resulting in a tradeoff between optical loss and single-mode behavior, which hinders the development of high-performance, ultralow-loss linear processing blocks on a single layer. Here, we demonstrate a dual-layer photonic integration approach with two silicon-nitride platforms exhibiting ultralow optical losses, i.e., a few dB/m, and individually optimized to perform either nonlinear or linear processing tasks. We demonstrate the functionality of this approach by integrating a power-efficient microcomb with an arrayed waveguide grating demultiplexer to filter a few frequency comb lines in the same monolithically integrated chip. This approach can significantly improve the integration of linear and nonlinear optical elements on a chip and opens the way to the development of fully integrated processing of Kerr nonlinear sources.