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.


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