# Non-evil definition of a Kleisli object in a weak 2-category

Let $$t : a \to a$$ be a monad in a weak 2-category. According to nLab, the 1-dimensional universal property of a Kleisli object $$(f_t : a \to a_t, \lambda : f_t t \to f_t)$$ is that for any right $$t$$-module on $$x$$, $$(r : a \to x, \alpha : r t \to r)$$, there is a unique morphism $$g : a_t\to x$$ such that $$r = g f_t$$ and $$\alpha = g \cdot \lambda$$.

But, in the case of a weak category, for 1-cells we do not want equalities (as it would be "evil") but we want isomorphisms instead. This means that the equality $$r = g f_t$$ should be replaced by an isomorphism between $$r$$ and $$g f_t$$. Now, shouldn't this isomorphism be unique? Could this be proved using the 2-dimensional universal property of a Kleisli object? Or do we need to add a coherence axiom to the definition?

P.S. See this answer for a more detailed definition of a Kleisli object than the one in nLab.