nLab
two-variable adjunction

Idea

An adjunction of two variables is a straightforward generalization of both:

• the internal hom in a biclosed monoidal category and

• a $V$-enriched category having powers and copowers

by extracting the central pattern.

Definition

Let $C$, $D$ and $E$ be categories. An adjunction of two variables or two-variable adjunction

$$(\otimes ,{\mathrm{hom}}_l,{\mathrm{hom}}_r):C\times D\to E$$(\otimes, hom_l, hom_r) : C \times D \to E

consists of functors

$$\begin{array}{rl}\otimes & :C\times D\to E\\ {\mathrm{hom}}_l& :C^{\mathrm{op}}\times E\to D\\ {\mathrm{hom}}_r& :D^{\mathrm{op}}\times E\to C\end{array}$$\begin{aligned} \otimes & : C \times D \to E \\ hom_l &: C^{op} \times E \to D \\ hom_r &: D^{op} \times E \to C \end{aligned}

together with natural isomorphisms

$${\mathrm{Hom}}_E(C\otimes D,E)\simeq {\mathrm{Hom}}_C(D,{\mathrm{hom}}_l(C,E))\simeq {\mathrm{Hom}}_D(C,{\mathrm{hom}}_r(D,E)).$$Hom_E(C \otimes D, E) \simeq Hom_C(D, hom_l(C,E)) \simeq Hom_D(C, hom_r(D,E)) \,.

References

This is in chapter 4 of

• Mark Hovey. Model Categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, 1999.