Adjoints of coarguments behave inconsistently
- Dominant language
- Python
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Description
Consider the following code (Firedrake syntax, but the issue is UFL):
```python3
from firedrake import *
m = UnitIntervalMesh(2)
V = FunctionSpace(m, "CG", 1)
ca = Coargument(V.dual(), 0)
```
`ca` is really the identity operator in $V^{\ast}\rightarrow V^{\ast}$ (equivalently it is a form in $V^{*}\times V \rightarrow R$) and it can arise in adjoint calculations when a variational problem has a pre-assembled cofunction in its RHS. The issue arises when taking the adjoint of `ca`. Note:
```
In [10]: adjoint(ca)
Out[10]: Argument(WithGeometry(FunctionSpace(, FiniteElement('Lagrange', interval, 1), name=None), Mesh(VectorElement(FiniteElement('Lagrange', interval, 1), dim=1), 0)), 0, None)
```
```
In [12]: adjoint(2*ca)
Out[12]: FormSum([2*Argument(WithGeometry(FunctionSpace(, FiniteElement('Lagrange', interval, 1), name=None), Mesh(VectorElement(FiniteElement('Lagrange', interval, 1), dim=1), 0)), 0, None)])
```
In effect, the adjoint of `ca` is treated as an unknown Function in `V` but the adjoint of a scalar multiple of `ca` is treated as a form in $V^{**}\times V^{*}\rightarrow R$ (equivalently an operator in $V\rightarrow V$). These spaces are mathematically isomorphic, but as a matter of type the former is a UFL expression while the latter is a form. When taking the adjoint of a form we expect to get back another form, so the latter would be preferable. This currently causes some special casing in Firedrake.
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