Item 8 (0191) — The Clowder Project

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

Monomorphisms and Epimorphisms. Let $\alpha \colon F\Longrightarrow G$ be a morphism of $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$. The following conditions are equivalent:
1. The natural transformation
  \[ \alpha \colon F \Longrightarrow G \]
  
  is a monomorphism (resp. epimorphism) in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$.
2. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the morphism
  \[ \alpha _{A} \colon F_{A} \to G_{A} \]
  
  is a monomorphism (resp. epimorphism) in $\mathcal{D}$.

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