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\homework{Limits}

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\maketitle

\begin{exercise}
Prove that for any object $\ob{A}$ of any category~$\cat{C}$, the object $\ob{A} \with \top$ (if it exists) is isomoprhic to $\ob{A}$.
\end{exercise}

\begin{exercise}
Prove that, in any 2-category, if morphisms $\ob{C}_1 \xmto{\mo{f}_1} \ob{C}_2$ and $\ob{C}_2 \xmto{\mo{f}_2} \ob{C}_3$ are both left adjoints, then their composition $\mo{f}_1 \cocomp \mo{f}_2$ is also a left adjoint.
\end{exercise}

\begin{exercise}
The monoid $\alg{A} \with \alg{B}$ is commutative if both $\alg{A}$ and $\alg{B}$ are commutative, and in that case is (with the appropriate projection homomorphisms) also the product of $\alg{A}$ and $\alg{B}$ in $\cat{CommMon}$.
Prove that there are morphisms $\kappa_\alg{A}$ and $\kappa_\alg{B}$ demonstrating that $\alg{A} \with \alg{B}$ is also the coproduct of $\alg{A}$ and $\alg{B}$ in $\cat{CommMon}$.
That is, prove that $\cat{CommMon}$ has \emph{biproducts}, meaning it has products and coproducts and they coincide on objects.
\end{exercise}

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