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closed monoidal category
Let $\mathcal{C}$ be a monoidal category, with tensor product $\otimes$. Then we say that

$\mathcal{C}$ is closed, or left closed, if the functor $A\otimes$ on $\mathcal{C}$ has a right adjoint $[A,]_{l}$

$\mathcal{C}$ is right closed if the functor $\otimes B$ on $\mathcal{C}$ has a right adjoint $[B,]_{r}$

$\mathcal{C}$ is biclosed if it is both left closed and right closed.
A biclosed symmetric monoidal category is also known as a symmetric monoidal closed category. In a symmetric monoidal closed category, $A\otimes B\cong B\otimes A$, so $[A,B]_{l}\cong[A,B]_{r}$. In this case, we denote the right adjoint by $[A,B]$.
Some examples:

Any cartesian closed category is symmetric monoidal closed.

An example of a biclosed monoidal category that is not symmetric monoidal is the category of bimodules over a noncommutative ring. The right adjoint of $A\times$ is $[A,]_{l}$, where $[A,B]_{l}$ is the collection of all left $R$linear bimodule homomorphisms from $A$ to $B$, while the right adjoint of $\times A$ is $[A,]_{r}$, where $[A,B]_{r}$ is the collection of all right $R$linear bimodule homomorphisms from $A$ to $B$. Unless $R$ is commutative, $[A,B]_{l}\neq[A,B]_{r}$ in general.
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