From my understanding after reading a few guides, ...

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[CLOVIS]

From my understanding after reading a few guides, Category theory is half Categories and half Morphisms

I understand that anything that can contain anything else is called a Functor, with multiple special cases (pointed functors...).

I forgot the name but there is one category that defines fold. The binary + and * operators define it. This category must be a morphism, right? If so, how is it called?