The limit of 0 is the terminal object
The limit of discrete categories $\geq 2$ are products.
But what is the limit of the discrete category with one object, 1?
2026-03-27 06:07:24.1774591644
Limit of 1 category
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A functor $F:\mathbf{1}\to C$ is determined by just a choice of an object $X$ of $C$ to send the one object of $\mathbf{1}$ to. The limit of $F$ is then just $X$ itself (with the identity map $X\to X$). I will leave it as an exercise to prove this has the required universal property.