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Let $F:\mathcal{C}\to\mathcal{D}$ be a functor between two categories $\mathcal{C}$ and $\mathcal{D}$ where the notion of $F$ preserving and commuting limits makes sense. I am unable to understand the difference between those two things. So my question is as follows:

What is the difference between a functor that commutes with limits and a functor that preserves limits?

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  • $\begingroup$ There is no difference. Both notions always make sense and are the same notion. $\endgroup$ – Qiaochu Yuan Feb 25 '15 at 5:04
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    $\begingroup$ @QiaochuYuan Oh, OK. Thanks! What about in the quasicategory case? Like what is the difference between "preserve small limits" and "commutes with finite limits"? $\endgroup$ – HiIloveMath Feb 25 '15 at 5:09
  • $\begingroup$ There is still no difference between "preserves" and "commutes with," but small limits are more general than finite limits. $\endgroup$ – Qiaochu Yuan Feb 25 '15 at 5:10
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    $\begingroup$ Because I wanted to make sure the OP was asking the question they intended to ask. $\endgroup$ – Qiaochu Yuan Feb 25 '15 at 5:47
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    $\begingroup$ Some people like to say "commutes", some people like to say "preserves". It's like saying a group homomorphism commutes with the group operation vs a group homomorphism preserves the group operation. $\endgroup$ – Zhen Lin Feb 25 '15 at 8:16
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In the comments it was claimed that the notions are equivalent. But I claim that there is a subtle difference in the language.

If $F : \mathcal{C} \to \mathcal{D}$ is a functor, then we say that $F$ preserves limits if for every limit cone $(L \to X_i)$ in $\mathcal{C}$ its image $(F(L) \to F(X_i))$ is also a limit cone in $\mathcal{D}$. Notice that this definition makes sense even if $\mathcal{C}$ has not all limits, and it is not assumed a priori that the diagram $(F(X_i))$ in $\mathcal{D}$ has some limit. But this would be necessary to speak of "commutation". Specifically, if $F : \mathcal{C} \to \mathcal{D}$ is a functor, $\mathcal{C}$ has limits of shape $I$ and $\mathcal{D}$ too, then the phrase "$F$ commutes with limits of shape $I$" should mean that the canonical morphism $$F(\lim_i X_i) \to \lim_i F(X_i)$$ is an isomorphism for every $I$-shaped diagram $X$ in $\mathcal{C}$. For $F$ to commute with some operation, we need the existence of such operation.

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