# stacks/stacks-project

 @@ -8554,17 +8554,55 @@ \section{Monoidal categories} \begin{remark} \label{remark-left-dual-adjoint} Lemma \ref{lemma-left-dual} says in particular that $Z \mapsto Z \otimes Y$ is the right adjoint of $Z' \mapsto Z' \otimes X$. Conversely, if this is true, then we get $\eta : \mathbf{1} \to X \otimes Y$ by evaluating the unit of the adjunction on $\mathbf{1}$ and $\epsilon : Y \otimes X \to \mathbf{1}$ by evaluating the counit of the adjunction on $\mathbf{1}$ (Section \ref{section-adjoint}). Thus the requirement that $Z \mapsto Z \otimes Y$ be the right adjoint of $Z' \mapsto Z' \otimes X$ is an equivalent formulation of the property of being a left dual. Uniqueness of adjoint functors guarantees that a left dual of $X$, if it exists, is unique up to unique isomorphism. is the right adjoint of $Z' \mapsto Z' \otimes X$. In particular, uniqueness of adjoint functors guarantees that a left dual of $X$, if it exists, is unique up to unique isomorphism. Conversely, assume the functor $Z \mapsto Z \otimes Y$ is a right adjoint of the functor $Z' \mapsto Z' \otimes X$, i.e., we're given a bijection $$\Mor(Z' \otimes X, Z) \longrightarrow \Mor(Z', Z \otimes Y)$$ functorial in both $Z$ and $Z'$. The unit of the adjunction produces maps $$\eta_Z : Z \to Z \otimes X \otimes Y$$ functorial in $Z$ and the counit of the adjoint produces maps $$\epsilon_{Z'} : Z' \otimes Y \otimes X \to Z'$$ functorial in $Z'$. In particular, we find $\eta = \eta_\mathbf{1} : \mathbf{1} \to X \otimes Y$ and $\epsilon = \epsilon_\mathbf{1} : Y \otimes X \to \mathbf{1}$. As an exercise in the relationship between units, counits, and the adjunction isomorphism, the reader can show that we have $$(\epsilon \otimes \text{id}_Y) \circ \eta_Y = \text{id}_Y \quad\text{and}\quad \epsilon_X \circ (\eta \otimes \text{id}_X) = \text{id}_X$$ However, this isn't enough to show that $(\epsilon \otimes \text{id}_Y) \circ (\text{id}_Y \otimes \eta) = \text{id}_Y$ and $(\text{id}_X \otimes \epsilon) \circ (\eta \otimes \text{id}_X) = \text{id}_X$, because we don't know in general that $\eta_Y = \text{id}_Y \otimes \eta$ and we don't know that $\epsilon_X = \epsilon \otimes \text{id}_X$. For this it would suffice to know that our adjunction isomorphism has the following property: for every $W, Z, Z'$ the diagram $$\xymatrix{ \Mor(Z' \otimes X, Z) \ar[r] \ar[d]_{\text{id}_W \otimes -} & \Mor(Z', Z \otimes Y) \ar[d]^{\text{id}_W \otimes -} \\ \Mor(W \otimes Z' \otimes X, W \otimes Z) \ar[r] & \Mor(W \otimes Z', W \otimes Z \otimes Y) }$$ If this holds, we will say {\it the adjunction is compatible with the given tensor structure}. Thus the requirement that $Z \mapsto Z \otimes Y$ be the right adjoint of $Z' \mapsto Z' \otimes X$ compatible with the given tensor structure is an equivalent formulation of the property of being a left dual. \end{remark} \begin{lemma}