# Multiplication and division by a morphism under the “inner composition” in closed monoidal categories

Let $${\mathcal C}$$ be a symmetric closed monoidal category, and let me denote the internal hom-functor by a fraction $$(X,Y)\mapsto\frac{Y}{X},$$ so that we have an isomorphism of functors $$\operatorname{Mor}(A\otimes B,C)\cong\operatorname{Mor}\left(A,\frac{C}{B}\right).$$ As is known, $${\mathcal C}$$ is an enriched category over itself. For each objects $$A,B,C$$ let me denote by $$\bullet_{A,B,C}$$ the "inner composition" in $${\mathcal C}$$ as in an enriched category, i.e. the morphism $$\bullet_{A,B,C}:\frac{C}{B}\otimes\frac{B}{A}\to\frac{C}{A}$$ with the corresponding properties.

I wonder if the following identity always holds $$\bullet_{A,C,D}\circ\left(1_{\frac{D}{C}}\otimes\frac{\varphi}{1_A}\right)= \bullet_{A,B,D}\circ\left(\frac{1_D}{\varphi}\otimes1_{\frac{B}{A}}\right)$$ (for arbitrary objects $$A,B,C,D$$ and for arbitrary morphism $$\varphi:B\to C$$). This is strange, I can prove this only in the case when the unit $$I$$ is a separating object in $${\mathcal C}$$ (what does not always hold). Is it possible that there is a couterexample?

Edit: I asked this now at mathoverflow.

• Where are you getting this "division" notation? – Derek Elkins Jun 23 at 0:59
• @DerekElkins this is my own attempt to shorten the writings. Fraction seems to be a convenient notation for closed monoidal categories: en.wikipedia.org/wiki/… – Sergei Akbarov Jun 23 at 4:52
• @ArnaudD. what do you think? – Sergei Akbarov Jun 23 at 9:51
• The division notation is actually rather popular among allegory theorists and you can even find it in category theory introductions such as Fokkinga's works. – Musa Al-hassy Jun 23 at 12:37
• Here's one of my favourites maartenfokkinga.github.io/utwente/mmf92b.pdf ;; it uses division for colimits ;; see "Categories, Allegories" for the use of division for residuals, which generalise Kan extensions ;-) – Musa Al-hassy Jun 26 at 13:14