Consider the functor $F: B \rightarrow Set$ where $B$ is a locally small category.
Is it true that if $F$ has a left adjoint then it is representable?
2026-02-22 21:49:51.1771796991
Is it true that a functor from a locally small category with a left adjoint is representable?
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Yes, it is. See for instance "Algebraic Theories: A Categorical Introduction to General Algebra" by J. Adámek, J. Rosický, E. M. Vitale, page 7 (chapter 0, section 0.10 about representable functors).
Proof: Let $L \dashv F$ and $1 = \{*\}$. Then, \begin{align} Fb \cong \mathbf{Set}(1,Fb) \cong B(L1,b) \end{align} naturally in $b \in B$, hence $F \cong H^{L1}$.