Let $\mathcal{K}$ be a $2$-category. Keeping in mind the Cat-intuition on $\mathcal{K}$ I say that:

Def : A $1$-cell $A \stackrel{f}{\to} B$ is a left split subobject if $f$ is a right adjoint and the counit is invertible.

In Cat this condition is equivalent to be fully faithful (and a right adjoint) for $f$.

In Lemma 2.3 of his 2-categories companion Steve Lack points out that:

In a 2-category, when the counit is invertible then $f$ is representably co - fully faithful.

Thus one implication remains true. Is it possible to have sort of a converse for this statement? Even adding somme additional structure on $\mathcal{K}$.

I was hoping to have something like the following:

A $1$-cell is a left split subobject if and only if $f$ is a right adjoint and a monomorphism in $\mathcal{K}$,

which - I understand - is a vain expectation. I was hoping to use the left cancellation in the triangle equality somehow. Unfortunately it is not working as I expected.