The equaliser of (f,g) is the pair consisting of:

  • The Limit. The pointed set (Eq(f,g),x0).
  • The Cone. The morphism of pointed sets

    eq(f,g):(Eq(f,g),x0)(X,x0)

    given by the canonical inclusion eq(f,g)Eq(f,g)X.

We claim that (Eq(f,g),x0) is the categorical equaliser of f and g in Sets. First we need to check that the relevant equaliser diagram commutes, i.e. that we have

feq(f,g)=geq(f,g),

which indeed holds by the definition of the set Eq(f,g). Next, we prove that Eq(f,g) satisfies the universal property of the equaliser. Suppose we have a diagram of the form

in Sets. Then there exists a unique morphism of pointed sets

ϕ:(E,)(Eq(f,g),x0)

making the diagram

commute, being uniquely determined by the condition

eq(f,g)ϕ=e

via

ϕ(x)=e(x)

for each xE, where we note that e(x)A indeed lies in Eq(f,g) by the condition

fe=ge,

which gives

f(e(x))=g(e(x))

for each xE, so that e(x)Eq(f,g). Lastly, we note that ϕ is indeed a morphism of pointed sets, as we have

ϕ()=e()=x0,

where we have used that e is a morphism of pointed sets.


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