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I'm looking for examples of the following situation:

Let $A$ be a complete and cocomplete category, $B$ is a small category and $T\colon B\to\mathbf{Set}^{A^{op}}$ be a functor, such that for any $b\in B$ functor $T(b)$ is representable. Could it be that the functor $\varprojlim T$ is not representable?

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By the Yoneda-Lemma, $T$ lifts to a diagram $X : B \to A$, which has a limit $\{L \to X_b\}$. Then one easily checks that $\{\hom(-,L) \to T_b\}$ is a limit (in fact the Yoneda embedding preserves all limits).

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  • $\begingroup$ Of course, I'm blind)) Thanks! $\endgroup$ – Oskar May 13 '13 at 18:25

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