Implication Details

Claim: If a category has cofiltered limits and has finite products, then it has products.

Proof: The product $\prod_{i \in I} X_i$ is the cofiltered limit of the finite partial products $\prod_{i \in E} X_i$ where $E$ ranges over the finite subsets of $I$ .

This implication has a dual.

Show 11 categories using this implication

category of finitely generated abelian groups
delooping of a group
delooping of an infinite uncountable group
category of finite abelian groups
category of finite groups
category of finite-dimensional vector spaces
category of finite-dimensional vector spaces [uncountable field]
partially ordered set of natural numbers
category of finitely generated projective modules over the ring of dual numbers
category of semigroups
category of combinatorial species