I am reading the paper of Pedersen: "Pullback and Pushout Constructions in C^*-Algebra Theory". I try to work out the arguments from Proposition $3.1$ of his paper (you can find this article in the world wide web). In particular the implication: "if pullback diagram then the three conditions hold. I know how to construct $\sigma:X\rightarrow A\oplus_CB$ and that by pullback property we get an unique arrow $\chi:A\oplus_CB\rightarrow X$. I can also prove that $\chi\circ\sigma=Id$ but with the other composition $\sigma\circ\chi=Id$ i have problems, because i have to show that this is true to conclude that $\sigma$ is an $*$-isomorphism. Someone an idea? Pedersen concludes that from this properties (i) and (ii) follows directly. Can you explain me how this follows from the fact that $\sigma$ is an isomorphism?
Thank you very much.