Item 1 (014C) — 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.

Characterisations. If $\mathcal{C}$ and $\mathcal{D}$ are small, then the following conditions are equivalent:
1. The functor $F$ is an isomorphism of categories.
2. The functor $F$ is fully faithful and bijective on objects.
3. For each $X\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
  \[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
  
  is an isomorphism of categories.
4. For each $X\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
  \[ F_{*}\colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]
  
  is an isomorphism of categories.

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