Suppose we have the following:

A C*-algebra $A$ and a von Neumann algebra $M$ (we can assume that $M$ is $\mathcal B(H)$).

A sequence of *-homomorphisms $\phi_i\colon A\to M$

an ultrafilter $\mathcal U\in\beta\mathbb N\setminus\mathbb N$

Define $\phi(a)=\lim_{i\in\mathcal U}\phi_i(a)$ for $a\in A$ where the limit is taken in the ultraweak topology of $M$.

Is $\phi$ a $*$-homomorphism?

If $A$ is unital and every $\phi_i$ is unital, it would be enough to have that $\phi$ is order zero, since then $\phi(1)=1$. So, is it true that given a sequence of cpc order zero maps from $A$ to $M$ and an ultrafilter, its point-ultraweak limit along the ultrafilter is order zero?

More generally, if we have two sequences of positive operators $S_i$ and $T_i$ such that $S_iT_i=0$ for every $i$, and $S=\lim_{\mathcal U}S_i$, $T=\lim_{\mathcal U}T_i$, are $S$ and $T$ orthogonal?