Is the notion of an adjunction well defined in an arbitrary weak $2$-category?

In particular, it seems like we need at least invertible unitors in the ambient $2$-category to write down the triangle identities. For a pair of antiparallel arrows $f:X\leftrightarrows Y:g$ in a $2$-category $\mathfrak{C}$ with unit $\eta:1_X\Rightarrow g\circ f$ and counit $\epsilon:f\circ g\Rightarrow1_Y$ we would like to write the triangle identities as follows

,

and as seen on the nLab entry for adjunctions, but in a weak $2$-category this diagram is incorrect. We've omitted the units and unitors:

In a partially weak $2$-category with invertible unitors this is okay, but in a fully weak $2$-category the unitors aren't invertible so the above diagrams are still wrong -- can we coherently define adjunctions inside a fully weak $2$-category?

At the $2$-categorical level we know that a fully weak $2$-category is equivalent to a fully strict one, and I suspect this allows for a workaround. At the $3$-categorical level we can only strictify $2$ of $3$ notions among associativity, units and interchange, so if a workaround exists at the $2$-categorical level we can probably strictify units in the equivalent weak $3$-category and repeat the solution one dimension up. At dimension $4$ and beyond we aren't certain about weakness being equivalent to strictness, so a workaround not using the strictification process would apply more generally with current technology.

Is the notion of an adjunction well defined in an arbitrary weak $2$-category?

In particular, it seems like we need at least invertible unitors in the ambient $2$-category to write down the triangle identities. For a pair of antiparallel arrows $f:X\leftrightarrows Y:g$ in a $2$-category $\mathfrak{C}$ with unit $\eta:1_X\Rightarrow g\circ f$ and counit $\epsilon:f\circ g\Rightarrow1_Y$ we would like to write the triangle identities as follows

,

and as seen on the nLab entry for adjunctions, but in a weak $2$-category this diagram is incorrect. We've omitted the units and unitors:

In a partially weak $2$-category with invertible unitors this is okay, but in a fully weak $2$-category the unitors aren't invertible so the above diagrams are still wrong -- can we coherently define adjunctions inside a fully weak $2$-category?

At the $2$-categorical level we know that a fully weak $2$-category is equivalent to a fully strict one, and I suspect this allows for a workaround. At the $3$-categorical level we can only strictify $2$ of $3$ notions among associativity, units and interchange, so if a workaround exists at the $2$-categorical level we can probably strictify units in the equivalent weak $3$-category and repeat the solution one dimension up. At dimension $4$ and beyond we aren't certain about weakness being equivalent to strictness, so a workaround not using the strictification process would apply more generally with current technology.

The above is incorrect, thanks to Todd and Kevin for pointing this out. I do not know if fully weak $2$-categories in the above sense are equivalent to their categories of presheaves under $2$-Yoneda or not.