Baby eczema or cat allergy?

Am

Just need an opinon of what y’all think this looks like. He developed this rash at 4 weeks old and he’s 5 weeks now.

It started out as small bumps like acne and then turned into this.

We took him to two hospitals they diagnosed it as baby acne and then at his one month appointment we showed the doctor and she diagnosed him with eczema.

Well my mother in law thinks he’s allergic to the cat and wants to get rid of our cat the reason she thinks this is because I’m somewhat allergic to cats and dogs and it might be hereditary.

Here’s the rash :

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