Item 9 (012Z) — 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.

For each $B\in \text{Obj}\webleft (\mathcal{D}\webright )$, there exist:
- An object $A_{B}$ of $\mathcal{C}$;
- A morphism $s_{B}\colon B\to F\webleft (A_{B}\webright )$ of $\mathcal{D}$;
- A morphism $r_{B}\colon F\webleft (A_{B}\webright )\to B$ of $\mathcal{D}$;
satisfying the following conditions:
1. The triple $\webleft (F\webleft (A_{B}\webright ),r_{B},s_{B}\webright )$ is a retract of $B$, i.e. we have $r_{B}\circ s_{B}=\text{id}_{B}$.
2. For each morphism $f\colon B'\to B$ of $\mathcal{D}$, we have
  \[ \webleft [\webleft (A_{B},s_{B'},f\circ r_{B'}\webright )\webright ]=\webleft [\webleft (A_{B},s_{B}\circ f,r_{B}\webright )\webright ] \]
  
  in $\int ^{A\in \mathcal{C}}h^{B'}_{F_{A}}\times h^{F_{A}}_{B}$.

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