Hi category theory stack exchange,
Is it true that the expression $(f;g)\otimes h=(f\otimes h);(g \otimes I)$ holds? It feels like it "should" from what I know about monoidal categories, but I am not sure how to assert it.
Here's a diagram to show what I mean:
$(f;g)\otimes h=(f\otimes h);(g \otimes I)$" />
I always felt like this was one of the "fundamental" properties of monoidal categories, the thing that makes them tick. But few of the monoidal category axioms actually address how the morphism $\times$ morphism $\rightarrow$ morphism aspect of the generating bifunctor behaves. There are lots of regularity criteria for the objects, but I have not found much for the morphisms. If anyone can clarify, it would be much appreciated!
The easiest way to check such questions is by drawing string diagrams, which make the answer absolutely immediate (the answer is "yes.") A good introduction to string diagrams is here.
That said, in this case, let $C$ be our monoidal category and let $f:x\to y,g:y\to z,h:u\to v.$ Then $(f;g)\otimes (h;\mathrm{id})$ is $\otimes((f,h);(g,\mathrm{id})),$ where now we write $\otimes$ explicitly as a functor $C\times C\to C.$ Then merely by functoriality, we conclude that $(f;g)\otimes (h;\mathrm{id})=\otimes((f,h);(g,\mathrm{id}))=\otimes((f;g,h))=(f;g)\otimes h,$ as we wanted to show.