Ken wrote:

>Def 12 on monoidal equivalence rubs me strangely, because FG->1 and GF->1 feel like they "want" to be a unit & counit

Yes, I was also waiting for an opportunity to ask about existence of identities similar to that of adjoints and what do they mean. Here are a few pictures:

![equiv](https://raw.githubusercontent.com/ikshv/categories/master/equiv.png?sanitize=true)

![coequiv](https://raw.githubusercontent.com/ikshv/categories/master/coequiv.png?sanitize=true)

So as I understand, **equivalence** means that we are trying to find a similar piece of structure between \\(C\\) and \\(D\\), and once we found it, we want some property being preserved. Still I need to catch up, so what I'm saying further is not precise at all, but it seems that with equivalence we are preserving products (or more generally limits). Also it seems that monoidal products can be constructed using products in a category. For example we can have a functor \\(\otimes: (b, c) \mapsto a, (b, a) \mapsto a, (c, a) \mapsto a\\), and using it we make \\(C\\) into monoidal category.

Similarly for co-equivalence (I totally made up this, need to find out what it name is), when you have natural transformation arrows reversed - this time we preserve coproducts.

From these one can easily pick up unit and counit. Using unit \\(1_C \Rightarrow G \circ F\\) we contract \\(C\\) to \\(G \circ F\\) along morphisms, and using counit \\(F \circ G \Rightarrow 1_D \\) we ensure that \\(G \circ F \circ G \circ F\\) is naturally isomorphic to \\(G \circ F\\).

Hopefully at least something from above makes any sense.

**EDIT 1** While I was writing this, John already answered this question in 11.

>Def 12 on monoidal equivalence rubs me strangely, because FG->1 and GF->1 feel like they "want" to be a unit & counit

Yes, I was also waiting for an opportunity to ask about existence of identities similar to that of adjoints and what do they mean. Here are a few pictures:

![equiv](https://raw.githubusercontent.com/ikshv/categories/master/equiv.png?sanitize=true)

![coequiv](https://raw.githubusercontent.com/ikshv/categories/master/coequiv.png?sanitize=true)

So as I understand, **equivalence** means that we are trying to find a similar piece of structure between \\(C\\) and \\(D\\), and once we found it, we want some property being preserved. Still I need to catch up, so what I'm saying further is not precise at all, but it seems that with equivalence we are preserving products (or more generally limits). Also it seems that monoidal products can be constructed using products in a category. For example we can have a functor \\(\otimes: (b, c) \mapsto a, (b, a) \mapsto a, (c, a) \mapsto a\\), and using it we make \\(C\\) into monoidal category.

Similarly for co-equivalence (I totally made up this, need to find out what it name is), when you have natural transformation arrows reversed - this time we preserve coproducts.

From these one can easily pick up unit and counit. Using unit \\(1_C \Rightarrow G \circ F\\) we contract \\(C\\) to \\(G \circ F\\) along morphisms, and using counit \\(F \circ G \Rightarrow 1_D \\) we ensure that \\(G \circ F \circ G \circ F\\) is naturally isomorphic to \\(G \circ F\\).

Hopefully at least something from above makes any sense.

**EDIT 1** While I was writing this, John already answered this question in 11.