**Grph**. It's tougher than I thought to get all the way there, but I see a few inroads. It's impressive how strong the assumption is latent in the word "closed". To assume, as is a consequence of assuming monoidal-closedness, that A ⊗ — preserves colimits gives you a lot!

For every object in

**Grph**can be gotten by taking some suitable colimit of the one-vertex graph

**1**and the one-edge graph

**2**. This means that I really only need to understand

**1**⊗

**1**,

**1**⊗

**2**,

**2**⊗

**2**

(and perhaps a few graph homomorphisms between them) to understand the action of ⊗ on the whole category.

Also, I know ⊗ has an identity object, I. Can I be the empty graph? No, because of preservation of colimits, G ⊗ 0 = 0, so that invalidates I being the identity if ever G is not zero. So now consider what I could possibly be. Whatever it is, it's the colimit of a bunch of copies of

**1**and

**2**. And observe that

**1**⊗ I =

**1**doesn't have any arrows in it. But again since ⊗ preserves colimits,

**1**⊗ I has to be the colimit of a bunch of copies of

**1**⊗

**1**and

**1**⊗

**2**. Weeelll, maybe there are no copies of

**1**⊗

**2**in there, since we haven't shown that I has any arrows, but at least there's a copy of

**1**⊗

**1**. Since colimits never get rid of arrows, only identify them with other arrows, we conclude that

a)

**1**⊗

**1**has no arrows in it.

b) Either

**1**⊗

**2**has no arrows, or I has no arrows (which would cause copies of

**1**⊗

**2**to show up in the colimit that's equal to

**1**⊗ I =

**1**)

Consider the case where I has no arrows. It's a collection of n vertices. But

**1**= I ⊗

**1**= n copies of

**1**, so n = 1, and

**1**= I. Therefore also we know the value of

**1**⊗

**2**: it's

**2**. Down this road is the non-cartesian tensor product that I described yesterday. All that remains is to somehow prove that

**2**⊗

**2**is a four-arrow square.

The case where

**1**⊗

**2**has no arrows ought to lead inexorably to the cartesian product, but I haven't made any further progress on that front.