I'm reading and struggling with bits and pieces of the book *Mal'cev, Protomodular, Homological, and Semi-Abelian categories* by Borceux and Bourn. At the moment I'm having trouble with:


**Theorem 1.3.22** Let $\mathsf C$ be a unital category. For all objects $X,Y\in \mathsf C$, the set $\boldsymbol{\mathsf Z}(X,Y)$ of central morphisms acts transitively on $\mathsf C(X,Y)$, where the action is given by addition.


The only bit of the proof (page 38) which seems relevant is the observation that given $z\in \boldsymbol{\mathsf Z}(X,Y),f\in \mathsf C(X,Y)$ the cooperator $\phi\_{z,f}$ is the composite $\phi \_z\circ (1\_X \times f)$. How does this show the action is transitive?


The transitivity of this action seems strange to me in concrete terms - it's saying every two arrows $X\rightrightarrows Y$ have a "difference" which is almost well defined, and I don't see what's going on here unless $\boldsymbol{\mathsf Z}(X,Y)$ is a group... What am I seeing wrong?




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**Added:**


**Definition.** A unital category is a finitely complete pointed category such that the unit injections $j\_i:A\_i\to A\_1\times A\_2$ given by $j\_1=(1,0),j\_2=(0,1)$ where $0,1$ are the zero and identity arrows are jointly extremally epimorphic. 


*Example.* The unital algebraic varieties are unital categories.


**Definition.** In a unital category, arrows $A\overset{f}{\to}C\overset{g}{\leftarrow}B$ are said to *cooperate* if there's a unique arrow $A\times B\to C$ making the obvious diagram with the $j\_i$ commute.


*Example.* Monoid homomorphisms cooperate iff their images commute pointwise, i.e $f(x)g(y)=g(y)f(x)$.


**Definition.** An arrow is central if it cooperates with the identity on its codomain.


*Example.* A submonoid is central iff its contained in the center of the monoid.



Why does the monoid of central morphisms act transitively?
The action is definitely not transitive. Consider for example the category of groups, which is unital. Here, the statement would be that for two groups $X,Y$ the group $\hom(X,Z(Y))$ (where $Z(Y)$ denotes the center of $Y$) acts transitively on the set $\hom(X,Y)$ via pointwise multiplication. But the orbit of the trivial homomorphism $X \to Y$ is just the set of homomorphisms $X \to Z(Y)$. Also, if $X=\mathbb{Z}$, we simply have the action of $Z(Y)$ on the underlying set of $Y$, whose orbit set can be quite large.


Perhaps the authors meant something different with transitive here, in particular because $\mathsf{Z}(X,Y)$ in general is just a commutative monoid?


