Let $C$ be a Small Category containing a Zero Object $Z$.
$(\forall{X,Y}\in(Obj(C)))$ define the Zero Morphism $\psi_{X,Y}:X\rightarrow{Y}$ as the composition $\psi_{X,Y}=I_{Y}\circ{T_{X}}$ where $T_{X}:X\rightarrow{Z}$ and $I_{Y}:Z\rightarrow{Y}$ are the Terminal and Initial morphisms respectively.
Define a Subcategory $C\left<Z\right>$ as follows:
- $Obj(C\left<Z\right>) =Obj(C)$
- $\forall{X,Y}\in{Obj(C)}$ $Hom_{C\left<Z\right>}(X,Y)$ $:={I_{Y}\circ{T_{X}}}$ i.e the only morphsisms are the Zero Isomorphisms.
I believe that $C\left<Z\right>$ forms a Connected Grouopoid, and that this construction should work in any Category containing a Zero Object, but I want to make sure I didn't miss anything.
There are two main flaws in your construction. Firstly, $C\left<Z\right>$ is not a category whenever it is not a groupoid, as identities of non-zero objects won't be included as morphisms. Secondly, if any arrow is an isomorphism in any subcategory of $C$, it must be also an isomorphism in $C$ itself, as all inverses existing in a subcategory are also morphisms in $C$. Thus again, only in the trivial case of groupoids it makes sense