Depends on their response to wavelengths at rest, including IR (for blueshift) and UV (for redshift).
Redshift doesn’t literally mean things appear red: the colors just remap, for example:
320 nm 🕶️ → 400 nm 🟪
400 nm 🟪 → 500 nm 🟩
500 nm 🟩 → 625 nm 🟧
625 nm 🟧 → 781 nm 💿 (IR, CD laser wavenength)
For easy calculation, I chose an exactly 25% increase in wavelength, achievable if the cat is receding at 25 % of the speed of light.
Or vice versa for blueshift. For example, a low-pressure sodium lamp in old street lights is near-monochromatic at 589 nm (pale yellow), which will reach 678nm red at 15% redshift, while a CD infrared laser incoming at the same 0.15𝑐 speed will have the same apparent red light at 678nm but because of blueshift.
TL;DR
🏳️🌈 ↓ Blueshift
🏳️🌈 ↑ Redshift
This is true about cats’ radiation but we usually mean reflectivity when describing a cat’s color. This of course means there is a light source illuminating the cat and the wavelengths observed also depend on the properties of the light source.