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I will refer to the notions recalled in

On Proposition $13.27$ of Adamek: full reflective subcategories are limits closed iff they are replete?

where I asked for clarifications about a result stated in "The Joy of Cats". The question I want to pose is the following: let $A$ be a (non necessarily replete) full subcategory of $B$. Is it true that the property of being closed under the formation of products and equalizers (or of products, pullbacks and terminal objects) is equivalent to that of being closed under the formation of all limits?

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  • $\begingroup$ If the containing category has all limits and the subcategory is closed under products and equalisers, then the subcategory is closed under limits. If the containing category does not have products and equalisers then the condition may be vacuous. $\endgroup$
    – Zhen Lin
    May 30, 2022 at 23:38
  • $\begingroup$ @ZhenLin I'm not convicted that such a result holds for non-replete subcategories. Can you provide me a bibliographic source where repleteness is not assumed? $\endgroup$ Jun 18, 2022 at 17:36
  • $\begingroup$ Repleteness is not really relevant. If the subcategory is replete you may have to replace some objects with isomorphic copies sometimes. This is of no significant consequence. $\endgroup$
    – Zhen Lin
    Jun 18, 2022 at 22:47

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