I'm trying to prove the following:
If $\mathcal{A}$ is an Abelian Category, then every object in $\mathcal{A}$ is projective if and only if every object is injective.
My attempt: Since every object in $\mathcal{A}$ is projective, we can find an epic morphism from any object to the others. Consequently, every object is isomorphic to 0 and so every object is injective. The converse is similar.
However, I don't think that my proof is true. Can anyone help?