= Finite-limit-preserving set-valued functor
= Left-exact set-valued functor
{synonym}
A <functor> $F:\mathcal C\to\mathbf{Set}$ is finite-limit-preserving if it carries each <finite limit> cone to a limiting cone. When $\mathcal C$ is a <small category> with <finite limits>, this is equivalent to $F$ being a <filtered colimit in a category> of covariant <representable functors>, and to the opposite of its <category of elements> being a <filtered category>. It is also equivalent to every <comma category> $(S\downarrow F)$ having <finite limits>, for all sets $S$.
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