--- > [!proposition]+ Proposition. ([[terminal objects are unique up to a unique isomorphism]]) > Let $\mathsf{C}$ be a [[category]]. >- If $I_{1},I_{2}$ are both [[terminal object|initial objects]] in $\mathsf{C}$, then $I_{1} \cong I_{2}$; >- If $F_{1},F_{2}$ are both [[terminal object|final objects]] in $\mathsf{C}$, then $F_{1} \cong F_{2}$. > Furthermore, these [[isomorphism|isomorphisms]] are uniquely determined. ^proposition > [!proof]+ Proof. ([[terminal objects are unique up to a unique isomorphism]]) > We'll treat both cases at once. Let $T_{1},T_{2}$ be either both initial or both final in $\mathsf{C}$. Then there is exactly one morphism $T_{1} \xrightarrow{f} T_{2}$ and exactly one morphism $T_{2} \xrightarrow{g}T_{1}$. We also have $\text{End}(T_{1})=\{ \id_{T_{1}} \}$ and $\text{End}(T_{2})=\{ \id_{T_{2}} \}$. Given that $gf \in \text{End}(T_{1})$ and $fg \in \text{End}(T_{2})$, it must be the case that $gf=\id_{T_{1}}$ and $fg=\id_{T_{2}}$. Thus in fact $f$ is an [[isomorphism]] with $f ^{-1}=g$, and because it is the only morphism from $T_{1}$ to $T_{2}$ it must be unique. ^proof --- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch02