Do we need braiding to define a monoid structure on product of monoids?

Let $(\mathcal{C}, \otimes, I)$ be a monoidal, symmetric category. It is well-known that in this case, if $M_{1}, M_{2}$ are monoids, there is a natural monoid structure on $M_{1} \otimes M_{1}$. The multiplication and unit are given as follows

$\phi: M_{1} \otimes M_{2} \otimes M_{1} \otimes M_{2} \rightarrow _{id \otimes \gamma \otimes id} M_{1} \otimes M_{1} \otimes M_{2} \otimes M_2 \rightarrow _{\phi _{1} \otimes \phi_{2}} \rightarrow M_{1} \otimes M_{2}$

$\eta: I \simeq I \otimes I \rightarrow _{\eta _{1} \otimes \eta_{2}} \rightarrow M_{1} \otimes M_{2}$

Do we actually need the symmetry of the monoidal category to obtain this construction? Is braiding enough? Is commutativity constraint enough?

Where can one find references for such facts, along with direct proofs? A nie introduction into the theory? Whenever I try to prove it on my own I have no idea how to use symmetry axioms to obtain associativity of the above multiplication. Yet, when I set, say, $\mathcal{C} = Mod(k)$, for a commutative ring $k$ and I write it down "element-wise", then the associativity is obvious!

I should say I'm not currently interested in symmetric monoidal categories per se (although I'm sure their theory is beautiful!), it's just whenever I go into a more complicated example of $\mathcal{C}$, like the category of graded modules (with braiding $a \otimes b \leadsto (-1)^{|a||b|} b \otimes a$) or, worse yet, chain complexes, I get immediately overwhelmed by signs and I lose the feeling of having total control over the objects in question. Are symmetric categories a way to go in this case? I imagine I could then just check the few axioms along with braiding and be 100% confident that everything else must work out.