This is a continuation from a previous question of mine about Logical disjunction in elementary toposes
I managed to prove conmutativity of both $\curlywedge$ and $\curlyvee$ (conjunction and disjunction as defined in Goldblatt's book Topoi) as well as asociativity of $\curlywedge$ by proving the relatively stronger result that $$\curlywedge\langle\pi_{1},\curlywedge\langle\pi_{2},\pi_{3}\rangle\rangle=\curlywedge\langle\pi_{\sigma(1)},\curlywedge\langle\pi_{\sigma(2)},\pi_{\sigma(3)}\rangle\rangle$$ for any permutation $\sigma\in\text{Sym}(3)$.
I did this by proving that $\curlywedge\langle\pi_{\sigma(1)},\curlywedge\langle\pi_{\sigma(2)},\pi_{\sigma(3)}\rangle\rangle$ is the character morphism of $\langle\top,\top,\top\rangle$ independently of $\sigma$, which is relatively easy.
Conmutativity for $\curlywedge$ can be proven using the universal property of the epi-mono factorization of $$w=[\langle\top_\Omega,\mathbf{1}_\Omega\rangle,\langle\mathbf{1}_\Omega,\top_\Omega\rangle]$$ and the fact that $$\langle\pi_{2},\pi_{1}\rangle w[i_{2},i_{1}]=w$$
However, disjunction is not easily generalizable as conjunction. I've tried similar ideas by finding the adequate $w$ for "three inputs", like $$w_3=[\langle\top_\Omega,\mathbf{1}_\Omega,\mathbf{1}_\Omega\rangle,\langle\mathbf{1}_\Omega,\top_\Omega,\mathbf{1}_\Omega\rangle,\langle\mathbf{1}_\Omega,\mathbf{1}_\Omega,\top_\Omega\rangle]$$ or $$w_3=[\langle\top_\Omega,\top_\Omega,\mathbf{1}_\Omega\rangle,\langle\top_\Omega,\mathbf{1}_\Omega,\top_\Omega\rangle,\langle\mathbf{1}_\Omega,\top_\Omega,\top_\Omega\rangle]$$
Nothing seems to work, i.e. I can't find appropiate morphsims $s,t$ such that $$sw_3t=w$$ so that I can use the universal property of the epi-mono factorization.
I'm convinced this must be the correct approach. Does someone know how to make it work?
Thanks in advance!