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If a functor $G:\mathcal{C}\rightarrow \mathcal{D}$ preserves all products and equalizers on pairs, is it continuous?

It seems like we can apply the same construction as the construction of arbitrary limits from products and equalizers in categories. In Mac Lane's Categories for the Working Mathematician Chapter 5 Section 4 Exercise 2, we seem to be given an extra assumption that $\mathcal{C}$ is complete, but I don't see why this is necessary.

VF1
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  • What is that exercise? – Berci Mar 25 '17 at 22:51
  • @Berci paraphrased, the exercises says: If $\mathcal{C}$ is complete, and $G:\mathcal{C}\rightarrow \mathcal{D}$ preserves products and equalizers, prove $G$ is continuous. – VF1 Mar 25 '17 at 23:35
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    If $\mathcal{C}$ is not complete, one has to wonder about limits in $\mathcal{C}$ that cannot be written as an equalizer of a product. –  Mar 26 '17 at 05:02
  • The statement is true if $\mathcal{C}$ has all products (because then each limit has the usual description as an equalizer of morphisms between products). In general it is false. (Actually, many people seem to believe that it is true, because they omit the assumption that $\mathcal{C}$ has products.) Perhaps I can find the counterexample ... – HeinrichD Mar 26 '17 at 10:11
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    It was discussed here two years ago, and a counterexample was mentioned (just dualize it): http://math.stackexchange.com/questions/1198452/ ... so perhaps this question can be closed as a duplicate. – HeinrichD Mar 26 '17 at 10:25
  • @HeinrichD would it also be OK if all we had was that the product representation in $\mathcal{D}$ of the limits existed (i.e., $\mathcal{D}$ has products)? – VF1 Mar 26 '17 at 13:25
  • @VF1: I don't understand the question. It is not enough that $\mathcal{D}$ has products (see the link; dualized). – HeinrichD Mar 26 '17 at 18:23
  • @HeinrichD You understood correctly; the dual of your linked example is indeed enough to answer both my questions in the negative. I suppose this is a duplicate, but only to category theorists... – VF1 Mar 27 '17 at 00:44

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