Implication details

Claim: If a category is retract-trivial, then it has reflexive coequalizers.

Proof: Let

f,g : X \rightrightarrows Y

be a reflexive pair, i.e. there is a morphism

r : Y \to X

with

f r = g r = \id_Y

Since

r

is a split monomorphism,

r

must be an isomorphism. Thus,

f = g = r^{-1}

and

\id_Y

is a coequalizer of

f,g

This implication has a dual.

Show 16 categories using this implication