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According to this question, the pullback in Cat exists by ordinary abstract nonsense. I've only built pullbacks explicitly in specific contexts such us the fibre product of schemes, pullback of sets and topological spaces... but these constructions required working on the specific category.

Where can I find the proof that they exist in Cat? For my purposes I only need it to exist in the subcategory of additive categories, but if it exists for all small categories then it's even better.

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    $\begingroup$ Regarding your last sentence : beware that existence of some limits in a category does not imply existence of limits in a subcategory, since the subcategory is not necessarily closed under limits, and could even have different limits ! $\endgroup$
    – Arnaud D.
    Jun 21, 2019 at 10:12
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    $\begingroup$ The linked question gives the construction of pullbacks of small categories. Have you tried just checking that it works? $\endgroup$
    – Arnaud D.
    Jun 21, 2019 at 10:14
  • $\begingroup$ @ArnaudD.okay, I see. Anyway, I'll be fine if they exists in at least one of those categories. $\endgroup$
    – Javi
    Jun 21, 2019 at 10:14
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    $\begingroup$ A small category is a model for an essentially algebraic theory. An essentially algebraic theory is just a small finitely complete category (or a presentation of such a category). The categories of models in $\mathcal E$ of an essentially algebraic theory are equivalent to categories of finite limit preserving $\mathcal E$-valued functors from the theory where $\mathcal E$ can be any category with finite limits but is usually $\mathbf{Set}$. The category of models in $\mathcal E$ will have any shapes of limits $\mathcal E$ has. This is very easy to prove. $\endgroup$ Jun 21, 2019 at 10:27
  • $\begingroup$ In the question you link they talk about both the 2-category $\mathfrak{Cat}$ and the 1-category (i.e. normal / usual category) $\mathbf{Cat}$. So when you ask about the category of (small) categories, do you mean the 2-category or the 1-category? $\endgroup$ Jun 21, 2019 at 13:58

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Cat is a bicomplete category i.e the category in which all small limits and colimits exist. (For reference regarding this fact, please check the examples of bicomplete categories in https://en.wikipedia.org/wiki/Complete_category) A pullback can be represented as a particular limit of a diagram. Hence 1-categorical Pullback exists in Cat.

Whereas when Cat is considered as a strict 2 category, Construction of the corresponding 2-categorical pullback in Cat is mentioned in https://stacks.math.columbia.edu/tag/003R

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    $\begingroup$ @jgon thanks for the comment. I have made the necessary edit in the answer. $\endgroup$ Mar 29, 2020 at 13:49

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