It's well-known that being cocartesian is a property of a *symmetric* monoidal category: $(\mathcal C,I,\otimes,\sigma)$ is cocartesian if and only if the following conditions are satisfied:

Every object $C \in \mathcal C$ has a commutative monoid structure $I \xrightarrow {u_C} C \overset{m_C}{\leftarrow} C \otimes C$;

With respect to the commutative monoid structures of (1), every morphism is a homomorphism (in other words, $u: const_I \Rightarrow 1_{\mathcal C}: \mathcal C \rightrightarrows \mathcal C$ and $m: \otimes \circ \Delta \Rightarrow 1_{\mathcal C}: \mathcal C \rightrightarrows \mathcal C$ are natural transformations -- here $\Delta: \mathcal C \to \mathcal C \times \mathcal C$ is the diagonal functor);

The commutative monoid structures of (1) satisfy $u_{C \otimes D} = u_C \otimes u_D$ and $m_{C \otimes D} = m_C \otimes m_D \circ (1_C \otimes \sigma_{D,C} \otimes 1_D)$ (in other words, $u$ and $m$ are monoidal natural transformations, where $\otimes$ has been made into a monoidal functor in the canonical way using the symmetry $\sigma$).

More precisely, if the data of (1,2,3) exists, then it is essentially unique, and exhibits $(\mathcal C, I, \otimes, \sigma)$ as cocartesian in the sense that $I$ is initial, and the maps $C \xrightarrow{1_C \otimes u_D} C \otimes D \overset{u_C \otimes 1_D}{\leftarrow} D$ are coproduct coprojections.

But what if $(\mathcal C, I,\otimes)$ is monoidal but not known to be symmetric monoidal?

**Questions:**

- Is the forgetful 2-functor from cocartesian monoidal categories to monoidal categories fully faithful?

I believe the answer to (1) is *yes*, as I'll sketch below. If this is correct, then "being cocartesian" is a *property* of a monoidal category. We can then ask:

- What is this property, explicitly?

**Partial progress:**

One might guess that the following conditions suffice:

Every object $C \in \mathcal C$ is a monoid object $(C,u_C,m_C)$.

Every morphism is a morphism of monoid objects.

Under these conditions, I can show that

The unit $I$ is the initial object;

There is a functorial construction of a morphism $C \otimes D \xrightarrow{m_E\circ(f\otimes g)} E$ from morphisms $f: C \to E, g: D \to E$, which restricts appropriately along the "candidate coproduct inclusions" $C \xrightarrow{1_C \otimes u_D} C \otimes D \overset{u_C \otimes 1_D}{\leftarrow} D$;

(If $\mathcal C$ has split idempotents) binary coproducts $C \amalg D$ exist, and are functorially split subobjects of $C \otimes D$, splitting the idempotent $m_{C \otimes D} \circ (1_C\otimes u_D \otimes u_C \otimes 1_D)$.

So I can show that $\mathcal C$ is cocartesian iff (1,2) hold and in addition the following condition is satisfied:

- For all $C,D \in \mathcal C$, $m_{C \otimes D} \circ(1_C \otimes u_D \otimes u_C \otimes 1_D) = 1_{C\otimes D}$

But I'm not quite sure how to motivate (3), nor am I sure whether it is superfluous.