Hi Nikos,
1. Suppose that \({\rm{I}}_1\),\({\rm{I}}_2\) are initial object in a category \(\mathcal{C}\). Because \({\rm{I}}_1\) is initial, exists unique morfism \({\rm{I}}_1\xrightarrow{f_1}{\rm{I}}_2\) and similarly, because \({\rm{I}}_2\) is initial, exists unique morfism \({\rm{I}}_2\xrightarrow{f_2}{\rm{I}}_1\). But then the morphisms \({\rm{I}}_1\xrightarrow{f_2\circ f_1}{\rm{I}}_1\) , \({\rm{I}}_2\xrightarrow{f_1\circ f_2}{\rm{I}}_2\) must be unique also. So \(f_2\circ f_1=id_{{\rm{I}}_1}\), \(f_1\circ f_2=id_{{\rm{I}}_2}\) and \({\rm{I}}_1\), \({\rm{I}}_2\) are equal up to isomorphism.
2. For the second, I assume that you are asking to prove the uniqueness of the zero object (when exists one). Is this correct? What do you define as zero object?
_________________ Grigorios Kostakos
