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Let $L:\mathscr A \times \mathscr B \longrightarrow \mathscr C$ and $R_1:\mathscr B^{op} \times \mathscr C \longrightarrow \mathscr A$ be two functors such that there is a bijection

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) $$ natural in $A,B,C$.

Is there any sufficient conditions to ensure existence of a functor $ R_2: \mathscr A^{op} \times \mathscr C \longrightarrow \mathscr B$ such that $(L,R_1,R_2)$ is an adjunction of two variables?

This mean that we would have bijections

$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R_1(B,C)) \cong \mathscr B( B, R_2(A,C)) $$ natural in $A,B,C$.

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up vote 4 down vote accepted

Firstly, note that it is enough to construct an isomorphism $$\mathcal{C}(L(A,B),C) \simeq \mathcal{B}(B, R_2(A,C))$$ The third natural isomorphism then follows automatically. Secondly, standard application of Yoneda's lemma shows that if for each $A\in \mathcal{A}$ you have a right adjoint $R_2(A,\cdot)$ to $L(A,\cdot)$, then these right adjoints are natural in $A$ and assemble into a bifunctor $R_2(A,C)$. There are various standard theorems that can be used to verify existence of right adjoint to each $L(A,\cdot)$, like Special Adjoint Functor Theorem or Freyd's AFT. The exact statement depends on your categories and functors. The simplest statement is Freyd's AFT: $\mathcal{B}$ must be cocomplete and locally small, $L(A,\cdot)$ must preserve all small colimits and satisfy solution set condition.

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I think you mean $A \in \mathscr A$, but anyway, seems good. Thanks for your answer. – Dimitri Zaganidis Mar 21 '13 at 17:26
Thank you, corrected. – Anton Fetisov Mar 21 '13 at 21:26

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