Let $\mathbf{B}$ and $\mathbf{B'}$ be strict bicategories and $F: \mathbf{B} \to \mathbf{B'}$ a weak functor which preserves horizontal composition strictly (i.e. $Ff * Fg = F(f * g)$ natural in f and g.)

Does this imply $F$ preserves identities strictly, i.e. $1_{F_a} = F(1_a)$?

With the unit axiom for weak functors, the strict preservation of $*$ and the strictness of $\mathbf{B}$ and $\mathbf{B'}$ it follows that

$$ Ff * 1_{Fa} = Ff = F(f * 1_a) = Ff * F1_a, $$

for arbitrary f. But I don't see how this implies $1_{F_a} = F(1_a).$

But if this isn't true, the claim that a cubical functor (see for instance definition 2 in https://arxiv.org/pdf/1409.2148.pdf ) automatically strictly preserves units would be false. (I've seen this claim as an aside now on at least 3 different occasions, so I might just miss something really simple here.)