Question about $\mbox{Ext}$ groups in abelian categories

Question

I'm studying basic properties of cotorsion theories in abelian categories. I am stuck in the proof of one assertion (not important which one exactly), which at some point claims that for any extension class $\xi \in \mbox{Ext}_\mathcal{C}^i(A,B)$ there exists a monomorphism $B \longrightarrow X$ such that the image of $\xi$ in $\mbox{Ext}_\mathcal{C}^i(A,X)$ vanishes. it's specified that this always holds by the definition of $\mbox{Ext}$ in abelian categories, but I cannot see why.

Answer 1

If your abelian category has enough injectives then you can take $B\to X$ to be a monomorphism from $B$ to an injective.

Or for any abelian category, using the Yoneda definition of $\text{Ext}^i$ as equivalence classes of $i$-fold extensions, then if $\zeta$ corresponds to the extension $$0\to B\to X_i\to\dots\to X_1\to A\to0$$ you can take the monomorphism $B\to X_i$.