Hi Sam, and thanks for requesting me to help with your question.
I'm sorry your five year old kitty is not feeling well.
From your description of the symptoms, there are various possibilities. If your older cat did not take well to the introduction of the new kitten, and then when your friend looked after them while you were traveling and allowed them to interact together, it's possible that stress has caused his inappropriate elimination (but the kitten, you said, had diarrhea, too), or if this is an intestinal infection, ingestion of something they shouldn't have eaten, or an intestinal parasite (not just worms), it most likely needs medication to resolve.
The fact that your cat's 3rd eyelid has become visible, is an indication that there is general illness in the body, so a vet visit for a hands-on exam and a fecal analysis would be a good next step.
For the moment, you can try to help the loose stools in both kitties by using Fortiflora for cats, mixed in with their canned food. Here is more information:
Also, feed them only bland food, in small portions, like plain boiled skinless chicken breast mixed with well cooked plain white rice and/or plain jarred baby food chicken containing NO onion or garlic (Beechnut Stage 1 chicken and chicken broth is a good one).
Use new Feliway Mulicat plug in diffusers in the rooms they occupy most, to help the stress and anxiety of the new introduction and your recent absence:
Make sure your cat is still eating, drinking and urinating normally, and add another litterbox or two, if you have room.
It would be best to have him seen asap, so try to get him into the vet, tomorrow; however, if symptoms worsen, bring him to an ER vet center right away.
I hope all will be well with your furry boy!
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