Implication Details

Claim: If a category has binary products and has coreflexive equalizers, then it has equalizers.

Proof: If $f,g : X \rightrightarrows Y$ are two morphisms, we have a coreflexive pair $(\id_X,f), (\id_X,g) : X \rightrightarrows X \times Y$ . A morphism with codomain $X$ equalizes $f$ and $g$ if and only if it equalizes $(\id_X,f)$ and $(\id_X,g)$ . Thus, their equalizers agree.

Show 12 categories using this implication

category of coproducts of Euclidean spaces
category of finite sets of even cardinality
category of finite sets of odd cardinality
category of finite sets of cardinality a power of 3
category of finitely generated free modules over Z x Z
category of smooth manifolds
category of partially ordered sets without isolated points
category of finitely generated projective modules over the ring of dual numbers
category of sets and relations
forked commutative square
walking idempotent
walking parallel pair