I'm reading "Abstract and Concrete Categories" and, in the Chap. I (Definition 3.52 on Page 41), there's an ``ObjectFree'' 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 "ObjectFree" version of a 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. 


As the others have said, the objectfree definition can always define objects later and do everything normally. One might ask if there was a differentthannormal definition that was more arrowlike. For example: A morphism $f$ is "terminating" if, for every morphism $g$, there exists a unique morphism $h$ such that $h \circ g$ and $f$ have the same target. So, the target of $f$ is a terminal object, and $f$ itself is the unique projection from its source. But this seems like a superficial change to me. Maybe someone else knows something better? 

