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).