countable copower functor on sets

notation: $\IN \times (-)$ : $\Set$ → $\Set$
Source: category of sets
Target: category of sets
Right adjoint: $(-)^{\IN}$
Related functors: $+$ , $2(-)$ , $\id_{\Set}$

This functor maps a set $X$ to the product $\IN \times X$ , which can also be seen as the copower $\IN \otimes X = \coprod_{n \in \IN} X$ . It is an example of a polynomial functor.

Satisfied Properties

Assigned properties

preserves equalizers
is cofinitary
is conservative
is a left adjoint

Deduced properties

preserves coreflexive equalizers
preserves monomorphisms
is faithful
is cocontinuous
is finitary
preserves coproducts
is right exact
is comonadic
preserves finite coproducts
preserves coequalizers
preserves epimorphisms
preserves initial objects
preserves reflexive coequalizers

Unsatisfied Properties

Assigned properties

does not preserve terminal objects
is not essentially surjective
is not essentially injective

Deduced properties*

is not an equivalence
does not preserve finite products
is not full
is not left-invertible
does not preserve products
is not left exact
is not continuous
is not exact
is not a right adjoint
is not representable
is not monadic

*This also uses the deduced satisfied properties.

Unknown properties

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