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38 Chapter 3. Schreier internal relations, categories and groupoids = qs0d1(σ) · σ · q(α) = σ · α.
It is easy to check that, when this formula holds, the composition becomes a monoid homomorphism. Moreover, when σ is in K[d1], we get
σ · α = q(s0d1(σ) · α) · σ = q(α) · σ = α · σ.
3.3 Schreier internal groupoids

An internal category X1 in a category E is a groupoid when, moreover, the following square determined by the composition map is a pullback:
X d1 //X 21
d0 X1
d0 X0,
// d0
or, in other words, when the following diagram is the kernel equivalence relation of d0:
d0 // d0 X2oos0 //X1 //X0
d1
or equivalently the natural comparison functor θ1 : Dec1X1 → R[d0] is an
isomorphism.
Definition 3.3.1. An internal groupoid in the category of monoids d0 //
is a Schreier internal groupoid if the split epimorphism (d0,s0) is a Schreier one. It is a homogeneous internal groupoid if (d0,s0) is homogeneous.
Again, as consequence of Proposition 2.3.3, we have that Schreier (resp. homogeneous) internal groupoids are closed under finite products inside the category of internal groupoids.
Proposition 3.3.2. Given a Schreier internal category X1, the following condi- tions are equivalent:
(a) X1 is a Schreier groupoid;
(b) R[d0] is a Schreier equivalence relation;
(c) K[d0] is a group.
X1oos0 //X0 d1