Fungal/eczema in nails

What product did you use? I do not see relation between products with no fragrance and alcohol and fungal infection since neither fragrance nor alcohol in skincare are used for antimicrobial properties. It's more likely to get fungal infection from someone else, if you share a product with someone (like nail polish) or from products with no preservatives that got contaminated. If you share nail accessories like nail scissors with someone and don't use disinfectants after someone then it's a way to get fungal infection. I would still recommend you to go to dermatologist since eczema can be an allergic reaction to substance - it might be a cosmetic substance but it can also be a detergent if you wash your dishes with your hands and don't wear protective gloves, etc

It’s unlikely that any of your skin care products are causing a fungal infection. Buy a product for nail fungus.