I have only just begun the Group Cohomology course, and my only background
is a Group Theory course so be gentle.

The Schur multiplier (M(G)) is defined as M(G)=A intersection X` , where
X->G is a projective central extension with kernel A (X and G are groups,
X->G is an epimorphism, and its kernel is a subgroup of Z(X), such that
for every other Y->H with the aforementioned properties, and for every
homomorphism G->H there exists a homomorphism X->Y such that the square is
commutative).
A very basic property of M(G) is that it is well-defined. Meaning it
doesn`t depend on the Y->H and G->H picked.
I have to mention that X->G is a projective central extension if and only
if X and G are groups, X->G is an epimorphism, and its kernel is a
subgroup of Z(X), such that for every other Y->G with the aforementioned
properties there exists a homomorphism X->Y such that the triangle is
commutative.
Somehow, using this property, they prove that M(G) is well-defined. I`m
afraid I can`t seem to prove it on my own... Can you help me?

Thanks ahead,
HH.