Definition 1.2.4 (Opposite Category).label Let $\frak{C}$ be a category then the oppposite category $\frak{C}^{\text{op}}$ is a category with the same objects with $\mor{\frak{C}^{\text{op}}}{X}{Y}\define \mor{\frak{C}}{Y}{X}$ and composition law
Then $\parens{\cd}^{\text{op}}$ is a strictly involutive contravariant endofunctor on $\text{CAT}_{U}$ as in Definition 1.2.5 and Definition 1.2.7, in particular $\parens{\frak{C}^{\text{op}}}^{\text{op}}=\frak{C}$ and $A\in \ob{\frak{C}}$ is intial (final) if and only if $A\in \ob{\frak{C}^{\text{op}}}$ is final (intial).
Proof. Observe
- (CAT1)
By construction
- (CAT2)
By construction
- (CAT3)
Let $X,Y,Z,W\in \ob{\frak{C}^{\text{op}}}$, $f\in \mor{\frak{C}^{\text{op}}}{X }{Y },g\in \mor{\frak{C}^{\text{op}}}{Y }{Z }$ and $h\in \mor{\frak{C}^{\text{op}}}{Z }{W }$ then
\begin{align*}(h\circ^{\text{op}}g)\circ^{\text{op}}f=(g\circ h)\circ^{\text{op}}f=f\circ(g\circ h)=(f\circ g)\circ h=h\circ^{\text{op}}(g\circ^{\text{op}}f)\end{align*}showing $\frak{C}^{\text{op}}$ is a category. The rest follows by construction.
$\square$
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