Is an algebra object $A$ in a monoidal category projective as a module over itself? Let $(\mathcal C, \otimes,e)$ be a monoidal category and $A$ an algebra object in $\mathcal C$. Denote $\mathcal C_A$ the category of $A$-right module objects in $\mathcal C$. 
My questions concern the projectivity of $A$.


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*Is $A$ projective as an $A$-right module object?

*Assuming this isn't true: Under which additional assumptions is $A$ projective as an $A$-right module.  For example if the tensor unit $e$ is projective in $\mathcal C$ this would hold but this seems to be a rather strong condition.

*Are there instructive (counter-) examples known?

 A: No, take $A = e$ to be the unit. The category of right $A$-modules is just $C$ again, so $e$ is projective as a right $e$-module iff $e$ is projective in $C$. So the unit being projective is necessary. 
Conversely, if the forgetful functor from right $A$-modules to $C$ has left adjoint given by $c \mapsto c \otimes A$ (I am not sure exactly what hypotheses are necessary for this; maybe it's always true), then the unit being projective implies that $A$ is projective as a right $A$-module, because left adjoints preserve projectives. (Edit, 9/11/19: This is false, whoops. What's true is that left adjoints preserve projectives if their right adjoints preserve epimorphisms.) 
For an explicit counterexample take $C$ to be the opposite of the monoidal category of abelian groups, also known as the category of compact Hausdorff abelian groups by Pontryagin duality. The unit object is $\mathbb{Z}^{op}$, or $S^1$, which is injective but not projective. Note that an algebra in $C$ is (the opposite of) a coalgebra in $\text{Ab}$ and a module over it is (the opposite of) a comodule in $\text{Ab}$, so your question reduces in this case to the question of whether a coalgebra is always injective as a right comodule over itself. 
For a counterexample which shows up "in nature" take $C$ to be the monoidal category of quasicoherent sheaves on any scheme $X$ which admits a quasicoherent sheaf with nontrivial sheaf cohomology; this is equivalent to the structure sheaf not being projective, and by Serre's criterion, if $X$ is quasicompact this is equivalent to $X$ not being affine. 
