Closed monoidal category

Remarks

If also every functor $B \otimes -$ has a right adjoint, $\mathcal{C}$ is called biclosed; for a symmetric monoidal category the two notions agree. Example: $\mathbf{Vect}_k$ with $[B, C]$ the space of linear maps $B \to C$; a linear map $f \colon A \otimes B \to C$ corresponds to $a \mapsto (b \mapsto f(a \otimes b))$. Non-example: $\mathbf{Set}$ with the disjoint union $\sqcup$ as tensor product and the empty set as unit is monoidal but not closed. A left adjoint preserves the initial object, but $\emptyset \sqcup B = B$ is not empty when $B$ is nonempty, so $- \sqcup B$ has no right adjoint. A Cartesian closed category is a closed monoidal category whose tensor product is the categorical product.