My kitten is jaundiced, lethargic, and won't eat. What's wrong?

Spartan may be suffering from a variety of different ailments. There are infectious disease, immune-mediated processes, and certain parasites known to cause the clinical signs that Spartan is exhibiting. If he is yellow as the pictures would suggest, his liver and biliary system are likely affected. He would most likely benefit from being in a hospitalized setting where he can receive fluid therapy and the appropriate treatments. Many times in cats showing these symptoms, there gastrointestinal tracts are not suited to digest / absorb oral medications, thereby making at-home therapy next to useless. While Synulox is an effective antibiotic, without more information and /or diagnostics, I would be using several additional medications beyond that. I would encourage you to get Spartan back into the vet on an emergency basis for reanalysis and reassessment.