All categories in this question are assumed to be small. Let $M$ be a symmetric closed monoidal category. I will write the composition maps for $M$-(enriched) categories by $\square_{a,b,c}:[a,b]\otimes[b,c]\to[a,c]$.

For $M$-categories $A,B$, define an $M$-category $A\otimes B$ by

  • $\operatorname{Ob}(A\otimes B) = \operatorname{Ob}(A)\times\operatorname{Ob}(B)$ (set-theoretic Cartesian product),
  • $[(a,b),(a'b')] = [a,a']\otimes[b,b']$,
  • $\boldsymbol{1}_{(a,b)} = \left(i\to i\otimes i\xrightarrow{\boldsymbol{1}_a\otimes\boldsymbol{1}_b}[a,a]\otimes[b,b]\right)$,
  • $\square_{(a,b),(a',b'),(a'',b'')} = \left(([a,a']\otimes[b,b'])\otimes([a',a'']\otimes[b',b''])\to([a,a']\otimes[a',a''])\otimes([b,b']\otimes[b',b''])\xrightarrow{\square\otimes\square}[a,a'']\otimes[b,b'']\right)$.

For $M$-functors $F:A\to C,\ G:B\to D$, define an $M$-functor $F\otimes G:A\otimes B\to C\otimes D$ by

  • $(F\otimes G)(a,b) = (F(a),G(b))$,
  • $(F\otimes G)_{(a,b),(a',b')} = F_{a,a'}\otimes G_{b,b'}$.

I want to prove that MCat, the category of $M$-categories, is a monoidal category. For that, I should define suitable associators and unitors in MCat. I think I know how to define associators:

  • $\alpha_{A,B,C}(a,(b,c)) = ((a,b),c)$,
  • $\left(\alpha_{A,B,C}\right)_{(a,(b,c)),(a',(b',c'))} = \alpha_{[a,a'],[b,b'],[c,c']}$.

But I do not know how to define unitors. I think $M$ should be the unity object of MCat. If I am right, then for each $M$-category $A$, a left unitor $\lambda_A$ should send each $(x,a)\in\operatorname{Ob}(M)\times\operatorname{Ob}(A)$ to an object in $A$, and send each $((x,a),(x',a'))$ to a morphism $[x,x']\otimes[a,a']\to[a,a']$. But I cannot think of an obvious such assignment.

  • 1
    $\begingroup$ No, the unit object $I$ of $M$-Cat is the category with a single object $*$, and where $I(*,*)$ is the monoidal unit of $M$. The unitors then are built out of the unitors of $M$, because $I\otimes A \cong A$ (obviously on objects, and $i\otimes A(x,y) \to A(x,y)$ is an isomorphism) $\endgroup$
    – fosco
    Jun 23 at 9:47


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