In Borceux's Handbook volume 1, page 145, the strong epis in the category of Banach spaces with bounded linear maps of norm <= 1 is characterized as the maps whose restriction on the unit balls is onto. However I don't follow the implication "strong epi" => "unitball unto". The argument given in the book is that if f: A ->> B is a strong epi, then for any element g: R -> B of B, the diagram
produces an arrow h as shown, and so f must be onto when restricted to the unitball. But this diagram is not the diagram of strong epi, which would be something like
where f is a strong epi, and z is monic, inducing an arrow w as shown.
What is it I'm not getting here?