I had a [similar confusion](https://forum.azimuthproject.org/discussion/comment/18678/#Comment_18678).

Consider three functions \\( L, C, R \\) where \\( C : A \rightarrow B \\) and \\( L, R : B \rightarrow A \\).

Where \\( L \\) and \\( R \\) are left and right adjoint to \\( C \\) respectively.

This implies that \\( C \\) is left adjoint to \\( R \\) and right adjoint to \\( L \\).

\\( R \\) is the approximate inverse of \\( C \\) from above and \\( L \\) is the approximate inverse of \\( C \\) from below. I am tempted to call them limits.

The error is in treating \\( L \\) and \\( R \\) as if they were left and right ajoints of each other, **they are not**.

What is the relationship between \\( L \\) and \\( R \\)?

It appears that they act as bounds for the true value in \\( C \\).

Consider three functions \\( L, C, R \\) where \\( C : A \rightarrow B \\) and \\( L, R : B \rightarrow A \\).

Where \\( L \\) and \\( R \\) are left and right adjoint to \\( C \\) respectively.

This implies that \\( C \\) is left adjoint to \\( R \\) and right adjoint to \\( L \\).

\\( R \\) is the approximate inverse of \\( C \\) from above and \\( L \\) is the approximate inverse of \\( C \\) from below. I am tempted to call them limits.

The error is in treating \\( L \\) and \\( R \\) as if they were left and right ajoints of each other, **they are not**.

What is the relationship between \\( L \\) and \\( R \\)?

It appears that they act as bounds for the true value in \\( C \\).