I read the article "The Homological Theory of maximal cohen-macaulay approximations" wrote by Auslander and Buchweitz. In this article in Lemma 3.1 in category theory drew commutative diagram by two exact sequence 
$$0 \rightarrow Y_A \rightarrow X_A \rightarrow A \rightarrow 0$$
such that above exact sequence is  X-approximaion of $A$ and 
$$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$
and since 
$$Ext^1(C,X_A) \rightarrow Ext^1(C,A)$$ 
is an isomorphism, there exist an exact commutative diagram like:
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/gu8qx.png

My questions are these: How to calculate Z? Is it a pull back or push out? Can anybody help me please? thanks

P.S: All the above were in the context of an abelian category, in which the existence of projective and/or injective objects is not necessary.