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86 Chapter 7. Special Schreier and special homogeneous surjections
horizontal relation. Since f¯ is the quotient of this relation, we get a unique factorization φˇ: Q → Hol(A) such that φˇf¯= φ¯.
Since the square (1) and the rectangle (1) + (2) are pullbacks, the square
(2) is also a pullback. According to the results of Section 5.2, we have that
the monoid Q is isomorphic to the semidirect product monoid A φ˜ Y , with
the operation given by (a,y) · (a′,y′) = (a · q(xR[f](x · a′)),y · y′), where x is
any element of X such that f(x) = y. Then the morphisms φˇ and f¯ can be
reinterpreted in the following way: φˇ: A φ˜ Y → Hol(A) is given by φˇ(a, y) = ¯¯′′
(a,φ˜(y))andf:R[f] A φ˜Y isgivenbyf(xR[f]x)=(q(xR[f]x),f(x)). 7.4 The direction functor in Mon
The previous construction of the direction actually gives rise to a functorial construction:
D: AbSExtY (Mon) → AbSPtY (Mon)
from the category of special Schreier surjections with codomain Y and abelian kernel to the abelian category AbSPtY (Mon) of internal abelian groups in the fiber SPtY (Mon) (which are nothing but Schreier split epimorphisms whose kernel is an abelian group). Any morphism u of special Schreier extensions:
A=K[f]//k //Xf////Y
K(u)

A′ =K[f′] //
k′
u

//X′ ////Y
f′
makes K(u) a Y -module homomorphism, and with it we associate the monoid homomorphism K(u) Y : A φ˜ Y → A′ ψ˜ Y with the respective appropriate classifying maps φ and ψ.
Proposition 7.4.1. The direction functor D preserves and reflects monomor- phisms, surjective homomorphisms and consequently isomorphisms.
Proof. It is a straightforward consequence of Proposition 7.2.2.
Before making explicit another extremely important property of the func- tor D, namely that it is cofibrant above the surjective homomorphisms, let us begin with the following immediate
Lemma 7.4.2. Let two Y -module structures φ: Y → End(A) and ψ: Y → End(A′) on the abelian groups A and A′ be given. A group homomorphism