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Stream: learning: questions

Topic: F-algebra homomorphism and naturality squares

Asad Saeeduddin (May 25 2020 at 16:56):

Is there any particular reason the commuting diagram associated with a morphism in the category of F-algebras looks so much like the naturality square for a natural transformation from F to the identity functor?

Morgan Rogers (he/him) (May 25 2020 at 17:04):

Is F something particular here or just any old endofunctor?

Jem (May 25 2020 at 18:12):

It is possible to express the property of being a morphism of algebras (over an endofunctor $F:C \to C$) in terms of natural transformations - an arrow $f:A \to B$ gives a morphism $f:(A,a) \to (B,b)$ iff $(a,b)$ give the components of a natural transformation $F\circ [f] \to [f]$, where $[f]:2\to C$ is the functor which picks out the arrow $f$.
I don't know if there's anything more interesting going on, though.