Yes, I phrased it in a wrong way. I meant "simplier", not "easier". The proof is really straightforward once you write down all the definitions. Thanks!

Thank you very much! I was reading "On Smooth Variational Principles in Banach Spaces", by Fabian, Hájek and Venderwerff and they reference the definition to justify this result, so I though it would be much easier. I see it's a bit more complicated. Again, thanks! This was really helpful.

Thank you very much! This is really helpful. I really appreaciate it!

If possible, could you elaborate? I'm not familiar with category theory language, but I really appreaciate your help!