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In page 1 of "Locally finitely presented additive categories", author says that a locally finitely presented category $\mathcal{A}$ is one for which every object can be expressed as a direct limit of finitely presented objects (with ignoring set-theoretic problems, and assuming that this category has direct limits). In page 2, he adds the condition "being skeletally small" for the full subcategory of finitely presented objects $f.p.(\mathcal A)$. Now, My question is: "Why do we need this assumption?" I cannot understand what happens if we omit that?

Remark: We say that an object $X$ in a category with direct limits is finitely presented if the functor $Hom(X,-)$ commutes with direct limits.

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Boris Chorny and Jiri Rosicky have a paper Locally class-presentable and locally class-accessible categories from 2012 where they generalize some of the results about locally accessible and locally presentable categories when the collection $A$ of finitely presented objects is not essentially small. As they say

The main disadvantage of $A$ being a class is that images of its objects by a functor can have arbitrarily large presentation ranks. Nevertheless, we will show that surprisingly many results about accessible categories can be generalized to the class-accessible setting. In particular, class-accessible categories are closed under lax limits. Furthermore, there is a satisfactory theory of injectivity and weak factorization systems in class-locally presentable categories which is a starting point for systematic applications in homotopy theory

You can read their paper to see what they generalize successfully. What does not generalize, however, is the following result (see https://ncatlab.org/nlab/show/adjoint+functor+theorem)

Theorem 2.2. Let $F\colon C\to D$ be a functor between locally presentable categories. Then

  • $F$ has a right adjoint if and only if it preserves all small colimits.
  • $F$ has a left adjoint if and only if it is an accessible functor and preserves all small limits.
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