From https://ncatlab.org/nlab/show/monoidal+category, a monoidal category requires a natural isomorphism
- a natural isomorphism $\lambda: (1 \otimes (-)) \rightarrow ^\cong (-)$ with components of the form $\lambda_x : 1 \otimes x \rightarrow x$
What does $(-)$ mean? Maybe the identity endofunctor? I suppose $(1 \otimes (-))$ is shorthand for a endofunctor which operates on objects by multiplying them by 1, but how does it operate on arrows?
Best Answer
Let $\mathscr{A}$ be a monoidal category. Let $A,A' \in \mathscr{A}$, and suppose $f \in \mathscr{A}(A,A')$.
You are right that $(-): \mathscr{A} \rightarrow \mathscr{A}$ is $\mathrm{id}_{\mathscr{A}}$.
$(1 \otimes (-)) : \mathscr{A} \rightarrow \mathscr{A}$ is defined as:
action on object $A$ given by $1 \otimes A$,
action on morphism $f$ given by $\mathrm{id}_1 \otimes f : 1 \otimes A \rightarrow 1 \otimes A'$.