I'm reading "Abstract and Concrete Categories" and, in the Chap. I
(Definition 3.52 on Page 41), there's an ``Object-Free'' definition of
a Category which, through the isomorphism $A \rightarrow \textit{id}\_A$
turns out to be equivalent to the usual one. But all the other
definitions in the book are only given in the usual "Objects+Morphisms"
fashion. What is the definition of Initial&Terminal Objects in the
"Object-Free" version of a Category?



Definition of Initial&Terminal Objects in an ``Object-Free'' Category
In this view, objects are equated with morphisms that are identities, or "units" in their terminology. So a morphism $x$ is initial when it is a unit and for every unit $y$ there is a unique morphism $f$ for which $f \circ x$ and $y \circ f$ are defined. Similarly, a morphism $y$ is terminal when it is a unit and for every unit $x$ there is a unique morphism $f$ for which $f \circ x$ and $y \circ f$ are defined.


