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This construction is used frequently (at least, I use it frequently in my work). For example, it appears in the usual proof that module categories have enough injectives. (In this case one studies $Cofree(\mathbb Q/\mathbb Z)$, as you anticipated.)

If we generalize slightly, and replace $\mathbb Z$ by the group ring $k[H]$ and $R$ by the group ring $k[G]$ (with $H$ being a subgroup of $G$), then $Hom_{k[H]}(k[G],\text{--})$ is precisely the functor of induction from $H$-representations to $G$-representations, and the adjointness you note is a form of Frobenius reciprocity.

If $R$ is a Hecke algebra (over $\mathbb Z$) on a space of weight $k$-cuspforms of some level, then $Cofree(\mathbb Z)$ is the space of modular forms of weight $k$ with coefficients in $\mathbb Z$. (This technical relationship between Hecke operators and the space of modular forms on which they operate is used frequently by number theorems theorists working on the arithmetic of modular forms.)

There are lots of other contexts in which this functor (and its variants, replacing $\mathbb Z$ by other rings) appear, but maybe I've said enough for now.

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This construction is used frequently (at least, I use it frequently in my work). For example, it appears in the usual proof that module categories have enough injectives. (In this case one studies $Cofree(\mathbb Q/\mathbb Z)$, as you anticipated.)

If we generalize slightly, and replace $\mathbb Z$ by the group ring $k[H]$ and $R$ by the group ring $k[G]$ (with $H$ being a subgroup of $G$), then $Hom_{k[H]}(k[G],\text{--})$ is precisely the functor of induction from $H$-representations to $G$-representations, and the adjointness you note is a form of Frobenius reciprocity.

If $R$ is a Hecke algebra (over $\mathbb Z$) on a space of weight $k$-cuspforms of some level, then $Cofree(\mathbb Z)$ is the space of modular forms of weight $k$ with coefficients in $\mathbb Z$. (This technical relationship between Hecke operators and the space of modular forms on which they operate is used frequently by number theorems working on the arithmetic of modular forms.)

There are lots of other contexts in which this functor (and its variants, replacing $\mathbb Z$ by other rings) appear, but maybe I've said enough for now.