Can the horizontal composition be treated as morphism product?
Let $A,B,C,D$ are functors such that compositions $A \circ C, B \circ D$ are suitable. $\alpha: A \to B, \beta: C \to D$ are 2 natural transformations.
Consider $A,B,C,D$ as objects and $\alpha, \beta$ as morphisms.
- Can the compositions $A \circ C, B \circ D$ be treated as the products of the corresponding objects (without any additional assumptions): $A \times C$ and $B \times D$? If not then is it possible if all of them are endofunctors of the same category?
- If yes then what is the product of the morphisms $\alpha \times \beta$? Is it the horizontal composition $\alpha \star \beta$?
EDIT: The original question was about relations of the following pairs of operations:
- $\times$ as product and $\times$ as morphisms product
- $\circ$ as functors composition and $\star$ as horizontal
composition
Both form bifunctors (aka tensor product) where the first operation is a mapping for objects and the second one can be treated as mapping for morphisms