In [Notions of Computation and Monads](https://core.ac.uk/download/pdf/21173011.pdf) Moggi models the notion of a computation of type $A$, $TA$, using a monad $T$. Among other things this ensures the $T\eta$ rule:

$$\frac{x: A \vdash a:TB}{x:A \vdash let \, y \Leftarrow a \, in\, [y] = a}$$

This rule prevents distinct computations having the same value. Is there a weakened notion of a monad that does not yield the $T\eta$ rule? Has this question already been studied? I'm particularly interested in the logic you get when you drop $T\eta$ from Moggi's rules for the simple metalanguage, but keep the other rules in place (table 4 in the pdf).