A monoidal category is left closed if for each $𝑌\in 𝐶$, the functor $-\otimes 𝑌:𝐶\to 𝐶$ has a right adjoint $𝑌⊸-:𝐶\to 𝐶$ forming the left internal-hom out of $𝑌$.

That is, for all $𝑋,𝑌,𝑍$, there is a natural isomorphism $𝐶[𝑋\otimes 𝑌,𝑍]\cong 𝐶[𝑋,[𝑌,𝑍]]$

There is an obvious right-handed variant $𝑌⟜-$ that is right adjoint to $𝑌\otimes -$.

If $𝐶$ is both right and left closed, the monoidal category is simply called closed (or perhaps biclosed).

A monoidal category is a category $\mathrm{𝒞︀}$ together with

1. a functor $\otimes :\mathrm{𝒞︀}\times \mathrm{𝒞︀}\to \mathrm{𝒞︀}$, the tensor product;
2. an object $𝐼$ of $\mathrm{𝒞︀}$, the unit;

3. natural isomorphisms

   $${𝛼}_{𝑥,𝑦,𝑧}:(𝑥\otimes 𝑦)\otimes 𝑧\to 𝑥\otimes (𝑦\otimes 𝑧){𝜆}_{𝑥}:𝐼\otimes 𝑥\to 𝑥{𝜌}_{𝑥}:𝑥\otimes 𝐼\to 𝑥$$

   the associator, left unitor and right unitor;

such that the triangle and the pentagon below commute for all objects $𝑤,𝑥,𝑦,𝑧$.