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Define a category as follows: the objects are categories and the arrows are embeddings (full and faithful functors). Is this category Cauchy complete or complete?

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It is easy to see that every idempotent splits in your category: if $e:C\to C$ is idempotent and fully faithful, let $B$ be the full subcategory of $C$ spanned by the objects in the image of $C$. Then the inclusion $B\to C$ is fully faithful, and $e$ factors fully faithfully through this inclusion, giving a splitting of $e$.

However, your category is not complete, because it does not have products. For instance, let $C$ be the discrete category with two objects $a$ and $b$. Suppose you have a product $P=C\times C$, with projection functors $p_0,p_1:P\to C$. There are two fully faithful functors $C\to C$, namely the identity $1$ and the functor $f$ which swaps $a$ and $b$. There is a unique fully faithful functor $g:C\to P$ such that $p_0g=p_1g=1$, and also a unique fully faithful functor $h:C\to P$ such that $p_0h=1$ and $p_1h=f$. Since the $p_i$ are fully faithful, $p_0g=p_0h=1$ implies $g(a)\cong h(a)$ and $g(b)\cong h(b)$ in $P$. But $p_1g=1$ and $p_1h=f$ implies $g(a)\cong h(b)$ and $g(b)\cong h(a)$. This implies $g(a)\cong g(b)$, contradicting that $g$ should be fully faithful.

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