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Conversely, given a groupoid G in the algebraic sense, let G < sub > 0 </ sub > be the set of all elements of the form x * x < sup >− 1 </ sup > with x varying through G and define G ( x * x < sup >-1 </ sup >, y * y < sup >-1 </ sup >) as the set of all elements f such that y * y < sup >-1 </ sup > * f * x * x < sup >-1 </ sup > exists.
Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).
To see this is well defined, observe that since z * z < sup >-1 </ sup > * g * y * y < sup >-1 </ sup > and y * y < sup >- 1 </ sup > * f * x * x < sup >- 1 </ sup > exist, so does z * z < sup >- 1 </ sup > * g * y * y < sup >- 1 </ sup > * y * y < sup >- 1 </ sup > * f * x * x < sup >-1 </ sup > = z * z < sup >- 1 </ sup > * g * f * x * x < sup >-1 </ sup >.
The identity morphism on x * x < sup >− 1 </ sup > is then x * x < sup >− 1 </ sup > itself, and the category-theoretic inverse of f is f < sup >- 1 </ sup >.

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